{"id":1716,"date":"2024-04-24T17:37:06","date_gmt":"2024-04-24T17:37:06","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus1\/?post_type=chapter&#038;p=1716"},"modified":"2025-08-18T00:08:29","modified_gmt":"2025-08-18T00:08:29","slug":"introduction-to-integration-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus1\/chapter\/introduction-to-integration-background-youll-need-1\/","title":{"raw":"Introduction to Integration: Background You'll Need 1","rendered":"Introduction to Integration: Background You&#8217;ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find the area of rectangles, triangles, trapezoids, and irregular shapes&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:769,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0}\">Find the area of rectangles, triangles, trapezoids, and irregular shapes<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Find the Area of a Rectangle<\/h2>\r\n<p>A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, [latex]L[\/latex], and the adjacent side as the width, [latex]W[\/latex].<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"189\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223837\/CNX_BMath_Figure_09_04_012.png\" alt=\"A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled L, the sides are labeled W.\" width=\"189\" height=\"123\" \/> Rectangle with all sides labeled[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The <strong>area of a rectangle<\/strong> is calculated as the product of its length and width. This relationship can be expressed through the formula:<\/p>\r\n<p style=\"text-align: center;\">[latex]A=L \\times W [\/latex]<\/p>\r\n<section class=\"textbox example\">\r\n<p>Consider a rectangular rug that is [latex]2[\/latex] feet long by [latex]3[\/latex] feet wide.<\/p>\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"241\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223838\/CNX_BMath_Figure_09_04_013.png\" alt=\"A rectangle made up of 6 squares. The bottom is 2 squares across and marked as 2, the side is 3 squares long and marked as 3.\" width=\"241\" height=\"178\" \/> A rug with length 3 and width 2, divided into unit squares.[\/caption]\r\n\r\n<p>The area of this rug would be:<\/p>\r\n<p style=\"text-align: center;\">[latex]A = 2 \\text{ ft } \\times 3 \\text{ ft } = 6 \\text{ square feet}[\/latex]<\/p>\r\n<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>area of rectangles<\/h3>\r\n<ul>\r\n\t<li>Rectangles have four sides and four right [latex]\\left(\\text{90}^ \\circ\\right)[\/latex] angles.<\/li>\r\n\t<li>The lengths of opposite sides are equal.<\/li>\r\n\t<li>The <strong>area<\/strong>, [latex]A[\/latex], of a rectangle is the length times the width. The area will be expressed in square units.<\/li>\r\n<\/ul>\r\n<center>[latex]A=L\\cdot W[\/latex]<\/center><\/div>\r\n<\/section>\r\n<section class=\"textbox example\">The length of a rectangle is [latex]32[\/latex] meters and the width is [latex]20[\/latex] meters. Find the area or the rectangle.<br \/>\r\n[reveal-answer q=\"172561\"]Show Solution[\/reveal-answer] [hidden-answer a=\"172561\"]\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td>\r\n[caption id=\"\" align=\"alignnone\" width=\"310\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223845\/CNX_BMath_Figure_09_04_068_img_MW-01.png\" alt=\"A rectangle with the top and bottom labeled 32 m and the sides labeled 20 m\" width=\"310\" height=\"176\" \/> Rectangle with all sides labeled[\/caption]\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the area of a rectangle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let [latex]A[\/latex] = the area<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula. Substitute.<\/td>\r\n<td>\r\n[caption id=\"\" align=\"alignnone\" width=\"310\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223846\/CNX_BMath_Figure_09_04_068_img_MW-02.png\" alt=\"The formula A = L times W. The formula is then written again with 32 substituted in for L and 20 substituted in for W\" width=\"310\" height=\"64\" \/> Formula for Area[\/caption]\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]A=640[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check.<\/strong><\/td>\r\n<td>\r\n<p>[latex]A\\stackrel{?}{=}640[\/latex]<\/p>\r\n<p>[latex]32\\cdot 20\\stackrel{?}{=}640[\/latex]<\/p>\r\n<p>[latex]640=640\\checkmark[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The area of the rectangle is [latex]640[\/latex] square meters.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]288389[\/ohm_question]<\/section>\r\n<section>\r\n<h2>Find the Area of a Triangle<\/h2>\r\n<p>We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle below, we\u2019ve labeled the length [latex]b[\/latex] and the width [latex]h[\/latex], so its area is [latex]bh[\/latex].<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"151\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223912\/CNX_BMath_Figure_09_04_035.png\" alt=\"A rectangle with the side labeled h and the bottom labeled b. The center says A equals bh.\" width=\"151\" height=\"89\" \/> Rectangle with height, base, and area labeled[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p><br \/>\r\nWe can divide this rectangle into two congruent triangles (see the image below). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or [latex]\\Large\\frac{1}{2}\\normalsize bh[\/latex]. This example helps us see why the formula for the area of a triangle is [latex]A=\\Large\\frac{1}{2}\\normalsize bh[\/latex].<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"323\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223913\/CNX_BMath_Figure_09_04_036.png\" alt=\"A rectangle with a diagonal line drawn from the upper left corner to the bottom right corner. The side of the rectangle is labeled h and the bottom is labeled b. Each triangle says one-half bh. To the right of the rectangle, it says &quot;Area of each triangle A = one-half bh&quot;. \" width=\"323\" height=\"107\" \/> Rectangle split into two triangles with height, base, and area labeled[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a [latex]\\text{90}^ \\circ[\/latex] angle with the base. The image below\u00a0shows three triangles with the base and height of each marked.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"563\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223914\/CNX_BMath_Figure_09_04_037.png\" alt=\"Three triangles. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h.\" width=\"563\" height=\"107\" \/> Examples of how the height of a triangle can be represented relative to its base[\/caption]\r\n<\/center>\r\n<section class=\"textbox keyTakeaway\">\r\n<h3>area of a triangle<\/h3>\r\n<p>The <strong>area<\/strong> of a triangle is one-half the base, [latex]b[\/latex], times the height, [latex]h[\/latex].<\/p>\r\n<p>&nbsp;<\/p>\r\n<p style=\"text-align: center;\">[latex]A={\\Large\\frac{1}{2}}bh[\/latex]<\/p>\r\n<p>&nbsp;<\/p>\r\n<br \/>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"190\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223917\/CNX_BMath_Figure_09_04_038_img.png\" alt=\"A triangle, with vertices labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.\" width=\"190\" height=\"160\" \/> Triangle with key features labeled[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<\/section>\r\n<section class=\"textbox example\">Find the area of a triangle whose base is [latex]11[\/latex] inches and whose height is [latex]8[\/latex] inches.<br \/>\r\n[reveal-answer q=\"247910\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"247910\"]&lt;tr\"&gt;Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.\r\n[caption id=\"\" align=\"alignnone\" width=\"318\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223918\/CNX_BMath_Figure_09_04_073_img-01.png\" alt=\"A triangle with the base labeled 11 in and a dotted vertical line from the top vertex to the base to form a right angle. This dotted line is labeled 8 in.\" width=\"318\" height=\"202\" \/> Triangle with height and base labeled[\/caption]\r\nStep 7. <strong>Answer<\/strong> the question.The area is [latex]44[\/latex] square inches.\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the area of the triangle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let [latex]A[\/latex] = area of the triangle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 4.<strong>Translate.<\/strong><\/p>\r\n<p>Write the appropriate formula.<\/p>\r\n<p>Substitute.<\/p>\r\n<\/td>\r\n<td style=\"height: 131px;\">\r\n[caption id=\"\" align=\"alignnone\" width=\"318\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223920\/CNX_BMath_Figure_09_04_073_img-02.png\" alt=\"The equation A = one half times b times h. The equation is written again with 11 substituted for b and 8 substituted for h.\" width=\"318\" height=\"110\" \/> Formula for Area[\/caption]\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]A=44[\/latex] square inches.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>\r\n<p>Step 6. <strong>Check.<\/strong><\/p>\r\n<\/td>\r\n<td>\r\n<p>[latex]A=\\frac{1}{2}bh[\/latex]<\/p>\r\n<p>[latex]44\\stackrel{?}{=}\\frac{1}{2}(11)8[\/latex]<\/p>\r\n<p>[latex]44=44\\quad\\checkmark[\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]288390[\/ohm_question]<\/p>\r\n<\/section>\r\n<h2>Find the Area of a Trapezoid<\/h2>\r\n<p>A <strong>trapezoid<\/strong> is four-sided figure, a <em>quadrilateral<\/em>, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base [latex]b[\/latex], and the length of the bigger base [latex]B[\/latex]. The height, [latex]h[\/latex], of a trapezoid is the distance between the two bases as shown in the image below.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"291\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223946\/CNX_BMath_Figure_09_04_052.png\" alt=\"A trapezoid, with the top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.\" width=\"291\" height=\"129\" \/> Trapezoid labeled with smaller base, larger base, and height[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The formula for the area of a trapezoid is: [latex]{\\text{Area}}_{\\text{trapezoid}}=\\Large\\frac{1}{2}\\normalsize h\\left(b+B\\right)[\/latex]. Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"179\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223948\/CNX_BMath_Figure_09_04_053.png\" alt=\"A trapezoid, with the top labeled with a small b and the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner.\" width=\"179\" height=\"123\" \/> Trapezoid split into two triangles with smaller base and larger base labeled[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The height of the trapezoid is also the height of each of the two triangles.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"193\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223949\/CNX_BMath_Figure_09_04_078.png\" alt=\"A trapezoid, with the top labeled with a small b and the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner. The upper right-hand side of the trapezoid forms a blue triangle, with the height of the trapezoid drawn in as a dotted line. The lower left-hand side of the trapezoid forms a red triangle, with the height of the trapezoid drawn in as a dotted line.\" width=\"193\" height=\"122\" \/> Trapezoid split into triangles and rectangles to illustrate its area formula[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>The formula for the area of a trapezoid is<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"185\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223950\/CNX_BMath_Figure_09_04_056_img.png\" alt=\"The formula for the area of a trapezoid, which is one half h times the quantity of lowercase b plus capital B\" width=\"185\" height=\"40\" \/> Formula for area of a trapezoid[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>If we distribute, we get,<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"201\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223951\/CNX_BMath_Figure_09_04_079_img.png\" alt=\"The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.\" width=\"201\" height=\"97\" \/> Formula for area of blue and red triangles[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>properties of trapezoids<\/h3>\r\n<ul id=\"fs-id1429217\">\r\n\t<li>A <strong>trapezoid<\/strong> has four sides.<\/li>\r\n\t<li>Two of its sides are parallel and two sides are not.<\/li>\r\n\t<li>The area, [latex]A[\/latex], of a trapezoid is [latex]\\text{A}=\\Large\\frac{1}{2}\\normalsize h\\left(b+B\\right)[\/latex] .<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<section class=\"textbox example\">Find the area of a trapezoid whose height is [latex]6[\/latex] inches and whose bases are [latex]14[\/latex] and [latex]11[\/latex] inches. [reveal-answer q=\"247911\"]Show Solution[\/reveal-answer] [hidden-answer a=\"247911\"]\r\n\r\n<table id=\"eip-id1168468452905\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td>\r\n[caption id=\"\" align=\"alignnone\" width=\"243\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223952\/CNX_BMath_Figure_09_04_080_img-01.png\" alt=\"A trapezoid with one parallel side labeled 11 in and the other labeled 14 in. There is a dotted line between the two parallel sides forming right angles with each of them. It is labeled 6 in.\" width=\"243\" height=\"148\" \/> Trapezoid with smaller base, larger base, and height labeled[\/caption]\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td>the area of the trapezoid<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td>Let [latex]A=\\text{the area}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4.<strong>Translate.<\/strong> Write the appropriate formula. Substitute.<\/td>\r\n<td>\r\n[caption id=\"\" align=\"alignnone\" width=\"392\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223953\/CNX_BMath_Figure_09_04_080_img-02.png\" alt=\"The equation A = one half times h times the quantity of little b plus big b. This formula is written again with 6 substituted in for h, 11 substituted in for little b and 14 substituted in for big b.\" width=\"392\" height=\"92\" \/> Formula for Area[\/caption]\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]A={\\Large\\frac{1}{2}}\\normalsize\\cdot 6(25)[\/latex] [latex]A=3(25)[\/latex] [latex]A=75[\/latex] square inches<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> Is this answer reasonable?<\/td>\r\n<td>\u00a0[latex]\\checkmark[\/latex]\u00a0 see reasoning below<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\nIf we draw a rectangle around the trapezoid that has the same big base [latex]B[\/latex] and a height [latex]h[\/latex], its area should be greater than that of the trapezoid. If we draw a rectangle inside the trapezoid that has the same little base [latex]b[\/latex] and a height [latex]h[\/latex], its area should be smaller than that of the trapezoid.<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"849\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223956\/CNX_BMath_Figure_09_04_060.png\" alt=\"A table is shown with 3 columns and 4 rows. The first column has an image of a trapezoid with a rectangle drawn around it in red. The larger base of the trapezoid is labeled 14 and is the same as the base of the rectangle. The height of the trapezoid is labeled 6 and is the same as the height of the rectangle. The smaller base of the trapezoid is labeled 11. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 14 times 6. Below is A sub rectangle equals 84 square inches. The second column has an image of a trapezoid. The larger base is labeled 14, the smaller base is labeled 11, and the height is labeled 6. Below this is A sub trapezoid equals one-half times h times parentheses little b plus big B. Below this is A sub trapezoid equals one-half times 6 times parentheses 11 plus 14. Below this is A sub trapezoid equals 75 square inches. The third column has an image of a trapezoid with a red rectangle drawn inside of it. The height is labeled 6. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 11 times 6. Below is A sub rectangle equals 66 square inches.\" width=\"849\" height=\"250\" \/> Using rectangle approximations to verify the trapezoid area formula[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n\r\nThe area of the larger rectangle is [latex]84[\/latex] square inches and the area of the smaller rectangle is [latex]66[\/latex] square inches. So it makes sense that the area of the trapezoid is between [latex]84[\/latex] and [latex]66[\/latex] square inches Step 7. <strong>Answer<\/strong> the question. The area of the trapezoid is [latex]75[\/latex] square inches. [\/hidden-answer]<\/section>\r\n<\/section>\r\n<section>\r\n<section class=\"textbox example\">\r\n<p>Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of [latex]3.4[\/latex] yards and the bases are [latex]8.2[\/latex] and [latex]5.6[\/latex] yards. How many square yards will be available to plant? [reveal-answer q=\"676574\"]Show Solution[\/reveal-answer] [hidden-answer a=\"676574\"]Solution<\/p>\r\n<table id=\"eip-id1168469766721\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Step 1 says, \">\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td style=\"width: 424.817px;\">\r\n[caption id=\"\" align=\"alignnone\" width=\"418\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224005\/CNX_BMath_Figure_09_04_082_img-01.png\" alt=\"A trapezoid with shorter base 5.6 yards and longer base 82 yards and a height of 3.4 yards.\" width=\"418\" height=\"129\" \/> Trapezoid with bases and height labeled[\/caption]\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\r\n<td style=\"width: 424.817px;\">the area of a trapezoid<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\r\n<td style=\"width: 424.817px;\">Let [latex]A[\/latex] = the area<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 4.<strong>Translate.<\/strong> Write the appropriate formula. Substitute.<\/td>\r\n<td style=\"width: 424.817px;\">\r\n[caption id=\"\" align=\"alignnone\" width=\"418\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224007\/CNX_BMath_Figure_09_04_082_img-02.png\" alt=\"The equation A = one half times h times the quantity of little b plus big b. The equation is rewritten with 3.4 substituted in for h, 5.6 substituted in for little b and 8.2 substituted in for big b.\" width=\"418\" height=\"101\" \/> Formular for Area[\/caption]\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td style=\"width: 424.817px;\">[latex]A={\\Large\\frac{1}{2}}\\normalsize(3.4)(13.8)[\/latex] [latex]A=23.46[\/latex] square yards.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 834px;\" colspan=\"2\">Step 6. <strong>Check:<\/strong> Is this answer reasonable? Yes. The area of the trapezoid is less than the area of a rectangle with a base of [latex]8.2[\/latex] yd and height [latex]3.4[\/latex] yd, but more than the area of a rectangle with base [latex]5.6[\/latex] yd and height [latex]3.4[\/latex] yd.<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"850\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224012\/CNX_BMath_Figure_09_04_066.png\" alt=\"A table with two rows. the first row is split into three columns. The first column is the formula Area of a rectangle (shown in red) equals base times height. On the next line under this it has numbers plugged into the formula; the base, 8.2 in parentheses times the height 3.4 in parentheses. Under this is it has the result 27.88 yards squared. The second column is the formula Area of a trapezoid with numbers already plugged in; one half times 3.4 yards times the quantity of 5.6 plus 8.2. Under this is has the result 23.46 yards squared. The third column is the formula Area of a rectangle (shown in blue) equals base times height. On the next line under it has number plugged into the formula; the base, 5.6 in parentheses times the height 3.4 in parentheses. Under this it has the result 19.04 yards squared. The second row shows that the Area of the red rectangle is greater than the Area of a trapezoid is greater than the Area of the blue rectangle. Beneath this, it shows the areas 27.88 for the red rectangle, 23.46 for the trapezoid, and 19.04 for the blue rectangle.\" width=\"850\" height=\"200\" \/> Comparison of areas of rectangles and a trapezoid with the same height[\/caption]\r\n<\/center><\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 409.183px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td style=\"width: 424.817px;\">Vinny has [latex]23.46[\/latex] square yards in which he can plant.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]146944[\/ohm_question]<\/p>\r\n<\/section>\r\n<h2>Find the Area of Irregular Figures<\/h2>\r\n<p>So far, we have found area for rectangles, triangles, and trapezoids. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. But some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.<\/p>\r\n<section class=\"textbox example\">Find the area of the shaded region.<br \/>\r\n<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"190\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224039\/CNX_BMath_Figure_09_05_012_img.png\" alt=\"An image of an attached horizontal rectangle and a vertical rectangle is shown. The top is labeled 12, the side of the horizontal rectangle is labeled 4. The side is labeled 10, the width of the vertical rectangle is labeled 2.\" width=\"190\" height=\"169\" \/> Shaded region with sides labeled[\/caption]\r\n<\/center><br \/>\r\n[reveal-answer q=\"247910\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"247910\"]<br \/>\r\nThe given figure is irregular, but we can break it into two rectangles. The area of the shaded region will be the sum of the areas of both rectangles.<br \/>\r\n<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"222\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224040\/CNX_BMath_Figure_09_05_013_img.png\" alt=\"An image of an attached horizontal rectangle and a vertical rectangle is shown. The top is labeled 12, the side of the horizontal rectangle is labeled 4. The side is labeled 10, the width of the vertical rectangle is labeled 2.\" width=\"222\" height=\"192\" \/> Shaded region split into two rectangles[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n\r\nThe blue rectangle has a width of [latex]12[\/latex] and a length of [latex]4[\/latex]. The red rectangle has a width of [latex]2[\/latex], but its length is not labeled. The right side of the figure is the length of the red rectangle plus the length of the blue rectangle. Since the right side of the blue rectangle is [latex]4[\/latex] units long, the length of the red rectangle must be [latex]6[\/latex] units.<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"215\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224041\/CNX_BMath_Figure_09_05_014_img.png\" alt=\"An image of a blue horizontal rectangle attached to a red vertical rectangle is shown. The top is labeled 12, the side of the blue rectangle is labeled 4. The whole side is labeled 10, the blue portion is labeled 4 and the red portion is labeled 6. The width of the red rectangle is labeled 2.\" width=\"215\" height=\"168\" \/> Shaded region split into two rectangles[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<br \/>\r\n<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"205\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224043\/CNX_BMath_Figure_09_05_015_img.png\" alt=\"The first line says A sub figure equals A sub rectangle plus A sub red rectangle. Below this is A sub figure equals bh plus red bh. Below this is A sub figure equals 12 times 4 plus red 2 times 6. Below this is A sub figure equals 48 plus red 12. Below this is A sub figure equals 60.\" width=\"205\" height=\"130\" \/> Formula for Area[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n\r\nThe area of the figure is [latex]60[\/latex] square units.<br \/>\r\n<br \/>\r\nIs there another way to split this figure into two rectangles? Try it, and make sure you get the same area.<br \/>\r\n[\/hidden-answer]<\/section>\r\n<\/section>\r\n<section>\r\n<section class=\"textbox example\">\r\n<p>Find the area of the shaded region.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"155\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224045\/CNX_BMath_Figure_09_05_018_img.png\" alt=\"A blue geometric shape is shown. It looks like a rectangle with a triangle attached to the top on the right side. The left side is labeled 4, the top 5, the bottom 8, the right side 7.\" width=\"155\" height=\"131\" \/> Shaded region with sides labeled[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>[reveal-answer q=\"937874\"]Show Solution[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"937874\"]<br \/>\r\nWe can break this irregular figure into a triangle and rectangle. The area of the figure will be the sum of the areas of the triangle and the rectangle. The rectangle has a length of [latex]8[\/latex] units and a width of [latex]4[\/latex] units. We need to find the base and height of the triangle.<br \/>\r\nSince both sides of the rectangle are [latex]4[\/latex], the vertical side of the triangle is [latex]3[\/latex] , which is [latex]7 - 4[\/latex] .<br \/>\r\nThe length of the rectangle is [latex]8[\/latex], so the base of the triangle will be [latex]3[\/latex] , which is [latex]8 - 5[\/latex] .<\/p>\r\n<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"178\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224046\/CNX_BMath_Figure_09_05_019_img.png\" alt=\"A geometric shape is shown. It is a blue rectangle with a red triangle attached to the top on the right side. The left side is labeled 4, the top 5, the bottom 8, the right side 7. The right side of the rectangle is labeled 4. The right side and bottom of the triangle are labeled 3.\" width=\"178\" height=\"131\" \/> Shaded region split into a triangle and a rectangle[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p><br \/>\r\nNow we can add the areas to find the area of the irregular figure.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"alignnone\" width=\"196\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224048\/CNX_BMath_Figure_09_05_020_img.png\" alt=\"The top line reads A sub figure equals A sub rectangle plus A sub red triangle. The second line reads A sub figure equals lw plus one-half red bh. The next line says A sub figure equals 8 times 4 plus one-half times red 3 times red 3. The next line reads A sub figure equals 32 plus red 4.5. The last line says A sub figure equals 36.5 sq. units.\" width=\"196\" height=\"169\" \/> Formula for Area[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p><br \/>\r\nThe area of the figure is [latex]36.5[\/latex] square units.[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">\r\n<p>[ohm_question hide_question_numbers=1]246488[\/ohm_question]<\/p>\r\n<\/section>\r\n<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-root=\"1\" data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Find the area of rectangles, triangles, trapezoids, and irregular shapes&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:769,&quot;3&quot;:{&quot;1&quot;:0},&quot;11&quot;:4,&quot;12&quot;:0}\">Find the area of rectangles, triangles, trapezoids, and irregular shapes<\/span><\/li>\n<\/ul>\n<\/section>\n<h2>Find the Area of a Rectangle<\/h2>\n<p>A rectangle has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, [latex]L[\/latex], and the adjacent side as the width, [latex]W[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 189px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223837\/CNX_BMath_Figure_09_04_012.png\" alt=\"A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled L, the sides are labeled W.\" width=\"189\" height=\"123\" \/><figcaption class=\"wp-caption-text\">Rectangle with all sides labeled<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The <strong>area of a rectangle<\/strong> is calculated as the product of its length and width. This relationship can be expressed through the formula:<\/p>\n<p style=\"text-align: center;\">[latex]A=L \\times W[\/latex]<\/p>\n<section class=\"textbox example\">\n<p>Consider a rectangular rug that is [latex]2[\/latex] feet long by [latex]3[\/latex] feet wide.<\/p>\n<figure style=\"width: 241px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223838\/CNX_BMath_Figure_09_04_013.png\" alt=\"A rectangle made up of 6 squares. The bottom is 2 squares across and marked as 2, the side is 3 squares long and marked as 3.\" width=\"241\" height=\"178\" \/><figcaption class=\"wp-caption-text\">A rug with length 3 and width 2, divided into unit squares.<\/figcaption><\/figure>\n<p>The area of this rug would be:<\/p>\n<p style=\"text-align: center;\">[latex]A = 2 \\text{ ft } \\times 3 \\text{ ft } = 6 \\text{ square feet}[\/latex]<\/p>\n<\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>area of rectangles<\/h3>\n<ul>\n<li>Rectangles have four sides and four right [latex]\\left(\\text{90}^ \\circ\\right)[\/latex] angles.<\/li>\n<li>The lengths of opposite sides are equal.<\/li>\n<li>The <strong>area<\/strong>, [latex]A[\/latex], of a rectangle is the length times the width. The area will be expressed in square units.<\/li>\n<\/ul>\n<div style=\"text-align: center;\">[latex]A=L\\cdot W[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox example\">The length of a rectangle is [latex]32[\/latex] meters and the width is [latex]20[\/latex] meters. Find the area or the rectangle.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q172561\">Show Solution<\/button> <\/p>\n<div id=\"q172561\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td>\n<figure style=\"width: 310px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223845\/CNX_BMath_Figure_09_04_068_img_MW-01.png\" alt=\"A rectangle with the top and bottom labeled 32 m and the sides labeled 20 m\" width=\"310\" height=\"176\" \/><figcaption class=\"wp-caption-text\">Rectangle with all sides labeled<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the area of a rectangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]A[\/latex] = the area<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula. Substitute.<\/td>\n<td>\n<figure style=\"width: 310px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223846\/CNX_BMath_Figure_09_04_068_img_MW-02.png\" alt=\"The formula A = L times W. The formula is then written again with 32 substituted in for L and 20 substituted in for W\" width=\"310\" height=\"64\" \/><figcaption class=\"wp-caption-text\">Formula for Area<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A=640[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>\n[latex]A\\stackrel{?}{=}640[\/latex]<br \/>\n[latex]32\\cdot 20\\stackrel{?}{=}640[\/latex]<br \/>\n[latex]640=640\\checkmark[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The area of the rectangle is [latex]640[\/latex] square meters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm288389\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288389&theme=lumen&iframe_resize_id=ohm288389&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section>\n<h2>Find the Area of a Triangle<\/h2>\n<p>We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle below, we\u2019ve labeled the length [latex]b[\/latex] and the width [latex]h[\/latex], so its area is [latex]bh[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 151px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223912\/CNX_BMath_Figure_09_04_035.png\" alt=\"A rectangle with the side labeled h and the bottom labeled b. The center says A equals bh.\" width=\"151\" height=\"89\" \/><figcaption class=\"wp-caption-text\">Rectangle with height, base, and area labeled<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>\nWe can divide this rectangle into two congruent triangles (see the image below). Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or [latex]\\Large\\frac{1}{2}\\normalsize bh[\/latex]. This example helps us see why the formula for the area of a triangle is [latex]A=\\Large\\frac{1}{2}\\normalsize bh[\/latex].<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 323px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223913\/CNX_BMath_Figure_09_04_036.png\" alt=\"A rectangle with a diagonal line drawn from the upper left corner to the bottom right corner. The side of the rectangle is labeled h and the bottom is labeled b. Each triangle says one-half bh. To the right of the rectangle, it says &quot;Area of each triangle A = one-half bh&quot;.\" width=\"323\" height=\"107\" \/><figcaption class=\"wp-caption-text\">Rectangle split into two triangles with height, base, and area labeled<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a [latex]\\text{90}^ \\circ[\/latex] angle with the base. The image below\u00a0shows three triangles with the base and height of each marked.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 563px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223914\/CNX_BMath_Figure_09_04_037.png\" alt=\"Three triangles. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h.\" width=\"563\" height=\"107\" \/><figcaption class=\"wp-caption-text\">Examples of how the height of a triangle can be represented relative to its base<\/figcaption><\/figure>\n<\/div>\n<section class=\"textbox keyTakeaway\">\n<h3>area of a triangle<\/h3>\n<p>The <strong>area<\/strong> of a triangle is one-half the base, [latex]b[\/latex], times the height, [latex]h[\/latex].<\/p>\n<p>&nbsp;<\/p>\n<p style=\"text-align: center;\">[latex]A={\\Large\\frac{1}{2}}bh[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div style=\"text-align: center;\">\n<figure style=\"width: 190px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223917\/CNX_BMath_Figure_09_04_038_img.png\" alt=\"A triangle, with vertices labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.\" width=\"190\" height=\"160\" \/><figcaption class=\"wp-caption-text\">Triangle with key features labeled<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<\/section>\n<section class=\"textbox example\">Find the area of a triangle whose base is [latex]11[\/latex] inches and whose height is [latex]8[\/latex] inches.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q247910\">Show Solution<\/button><\/p>\n<div id=\"q247910\" class=\"hidden-answer\" style=\"display: none\">&lt;tr&#8221;&gt;Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/p>\n<figure style=\"width: 318px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223918\/CNX_BMath_Figure_09_04_073_img-01.png\" alt=\"A triangle with the base labeled 11 in and a dotted vertical line from the top vertex to the base to form a right angle. This dotted line is labeled 8 in.\" width=\"318\" height=\"202\" \/><figcaption class=\"wp-caption-text\">Triangle with height and base labeled<\/figcaption><\/figure>\n<p>Step 7. <strong>Answer<\/strong> the question.The area is [latex]44[\/latex] square inches.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the area of the triangle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]A[\/latex] = area of the triangle<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 4.<strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula.<\/p>\n<p>Substitute.<\/p>\n<\/td>\n<td style=\"height: 131px;\">\n<figure style=\"width: 318px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223920\/CNX_BMath_Figure_09_04_073_img-02.png\" alt=\"The equation A = one half times b times h. The equation is written again with 11 substituted for b and 8 substituted for h.\" width=\"318\" height=\"110\" \/><figcaption class=\"wp-caption-text\">Formula for Area<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A=44[\/latex] square inches.<\/td>\n<\/tr>\n<tr>\n<td>\n<p>Step 6. <strong>Check.<\/strong><\/p>\n<\/td>\n<td>\n[latex]A=\\frac{1}{2}bh[\/latex]<br \/>\n[latex]44\\stackrel{?}{=}\\frac{1}{2}(11)8[\/latex]<br \/>\n[latex]44=44\\quad\\checkmark[\/latex]\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm288390\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=288390&theme=lumen&iframe_resize_id=ohm288390&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<h2>Find the Area of a Trapezoid<\/h2>\n<p>A <strong>trapezoid<\/strong> is four-sided figure, a <em>quadrilateral<\/em>, with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base [latex]b[\/latex], and the length of the bigger base [latex]B[\/latex]. The height, [latex]h[\/latex], of a trapezoid is the distance between the two bases as shown in the image below.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 291px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223946\/CNX_BMath_Figure_09_04_052.png\" alt=\"A trapezoid, with the top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.\" width=\"291\" height=\"129\" \/><figcaption class=\"wp-caption-text\">Trapezoid labeled with smaller base, larger base, and height<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The formula for the area of a trapezoid is: [latex]{\\text{Area}}_{\\text{trapezoid}}=\\Large\\frac{1}{2}\\normalsize h\\left(b+B\\right)[\/latex]. Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 179px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223948\/CNX_BMath_Figure_09_04_053.png\" alt=\"A trapezoid, with the top labeled with a small b and the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner.\" width=\"179\" height=\"123\" \/><figcaption class=\"wp-caption-text\">Trapezoid split into two triangles with smaller base and larger base labeled<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The height of the trapezoid is also the height of each of the two triangles.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 193px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223949\/CNX_BMath_Figure_09_04_078.png\" alt=\"A trapezoid, with the top labeled with a small b and the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner. The upper right-hand side of the trapezoid forms a blue triangle, with the height of the trapezoid drawn in as a dotted line. The lower left-hand side of the trapezoid forms a red triangle, with the height of the trapezoid drawn in as a dotted line.\" width=\"193\" height=\"122\" \/><figcaption class=\"wp-caption-text\">Trapezoid split into triangles and rectangles to illustrate its area formula<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The formula for the area of a trapezoid is<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 185px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223950\/CNX_BMath_Figure_09_04_056_img.png\" alt=\"The formula for the area of a trapezoid, which is one half h times the quantity of lowercase b plus capital B\" width=\"185\" height=\"40\" \/><figcaption class=\"wp-caption-text\">Formula for area of a trapezoid<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>If we distribute, we get,<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 201px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223951\/CNX_BMath_Figure_09_04_079_img.png\" alt=\"The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.\" width=\"201\" height=\"97\" \/><figcaption class=\"wp-caption-text\">Formula for area of blue and red triangles<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>properties of trapezoids<\/h3>\n<ul id=\"fs-id1429217\">\n<li>A <strong>trapezoid<\/strong> has four sides.<\/li>\n<li>Two of its sides are parallel and two sides are not.<\/li>\n<li>The area, [latex]A[\/latex], of a trapezoid is [latex]\\text{A}=\\Large\\frac{1}{2}\\normalsize h\\left(b+B\\right)[\/latex] .<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<section class=\"textbox example\">Find the area of a trapezoid whose height is [latex]6[\/latex] inches and whose bases are [latex]14[\/latex] and [latex]11[\/latex] inches. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q247911\">Show Solution<\/button> <\/p>\n<div id=\"q247911\" class=\"hidden-answer\" style=\"display: none\">\n<table id=\"eip-id1168468452905\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td>\n<figure style=\"width: 243px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223952\/CNX_BMath_Figure_09_04_080_img-01.png\" alt=\"A trapezoid with one parallel side labeled 11 in and the other labeled 14 in. There is a dotted line between the two parallel sides forming right angles with each of them. It is labeled 6 in.\" width=\"243\" height=\"148\" \/><figcaption class=\"wp-caption-text\">Trapezoid with smaller base, larger base, and height labeled<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the area of the trapezoid<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]A=\\text{the area}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 4.<strong>Translate.<\/strong> Write the appropriate formula. Substitute.<\/td>\n<td>\n<figure style=\"width: 392px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223953\/CNX_BMath_Figure_09_04_080_img-02.png\" alt=\"The equation A = one half times h times the quantity of little b plus big b. This formula is written again with 6 substituted in for h, 11 substituted in for little b and 14 substituted in for big b.\" width=\"392\" height=\"92\" \/><figcaption class=\"wp-caption-text\">Formula for Area<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A={\\Large\\frac{1}{2}}\\normalsize\\cdot 6(25)[\/latex] [latex]A=3(25)[\/latex] [latex]A=75[\/latex] square inches<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> Is this answer reasonable?<\/td>\n<td>\u00a0[latex]\\checkmark[\/latex]\u00a0 see reasoning below<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If we draw a rectangle around the trapezoid that has the same big base [latex]B[\/latex] and a height [latex]h[\/latex], its area should be greater than that of the trapezoid. If we draw a rectangle inside the trapezoid that has the same little base [latex]b[\/latex] and a height [latex]h[\/latex], its area should be smaller than that of the trapezoid.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 849px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24223956\/CNX_BMath_Figure_09_04_060.png\" alt=\"A table is shown with 3 columns and 4 rows. The first column has an image of a trapezoid with a rectangle drawn around it in red. The larger base of the trapezoid is labeled 14 and is the same as the base of the rectangle. The height of the trapezoid is labeled 6 and is the same as the height of the rectangle. The smaller base of the trapezoid is labeled 11. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 14 times 6. Below is A sub rectangle equals 84 square inches. The second column has an image of a trapezoid. The larger base is labeled 14, the smaller base is labeled 11, and the height is labeled 6. Below this is A sub trapezoid equals one-half times h times parentheses little b plus big B. Below this is A sub trapezoid equals one-half times 6 times parentheses 11 plus 14. Below this is A sub trapezoid equals 75 square inches. The third column has an image of a trapezoid with a red rectangle drawn inside of it. The height is labeled 6. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 11 times 6. Below is A sub rectangle equals 66 square inches.\" width=\"849\" height=\"250\" \/><figcaption class=\"wp-caption-text\">Using rectangle approximations to verify the trapezoid area formula<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The area of the larger rectangle is [latex]84[\/latex] square inches and the area of the smaller rectangle is [latex]66[\/latex] square inches. So it makes sense that the area of the trapezoid is between [latex]84[\/latex] and [latex]66[\/latex] square inches Step 7. <strong>Answer<\/strong> the question. The area of the trapezoid is [latex]75[\/latex] square inches. <\/div>\n<\/div>\n<\/section>\n<\/section>\n<section>\n<section class=\"textbox example\">\n<p>Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of [latex]3.4[\/latex] yards and the bases are [latex]8.2[\/latex] and [latex]5.6[\/latex] yards. How many square yards will be available to plant? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q676574\">Show Solution<\/button> <\/p>\n<div id=\"q676574\" class=\"hidden-answer\" style=\"display: none\">Solution<\/p>\n<table id=\"eip-id1168469766721\" class=\"unnumbered unstyled\" style=\"width: 859px;\" summary=\"Step 1 says,\">\n<tbody>\n<tr>\n<td style=\"width: 409.183px;\">Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td style=\"width: 424.817px;\">\n<figure style=\"width: 418px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224005\/CNX_BMath_Figure_09_04_082_img-01.png\" alt=\"A trapezoid with shorter base 5.6 yards and longer base 82 yards and a height of 3.4 yards.\" width=\"418\" height=\"129\" \/><figcaption class=\"wp-caption-text\">Trapezoid with bases and height labeled<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 409.183px;\">Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td style=\"width: 424.817px;\">the area of a trapezoid<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 409.183px;\">Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td style=\"width: 424.817px;\">Let [latex]A[\/latex] = the area<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 409.183px;\">Step 4.<strong>Translate.<\/strong> Write the appropriate formula. Substitute.<\/td>\n<td style=\"width: 424.817px;\">\n<figure style=\"width: 418px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224007\/CNX_BMath_Figure_09_04_082_img-02.png\" alt=\"The equation A = one half times h times the quantity of little b plus big b. The equation is rewritten with 3.4 substituted in for h, 5.6 substituted in for little b and 8.2 substituted in for big b.\" width=\"418\" height=\"101\" \/><figcaption class=\"wp-caption-text\">Formular for Area<\/figcaption><\/figure>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 409.183px;\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td style=\"width: 424.817px;\">[latex]A={\\Large\\frac{1}{2}}\\normalsize(3.4)(13.8)[\/latex] [latex]A=23.46[\/latex] square yards.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 834px;\" colspan=\"2\">Step 6. <strong>Check:<\/strong> Is this answer reasonable? Yes. The area of the trapezoid is less than the area of a rectangle with a base of [latex]8.2[\/latex] yd and height [latex]3.4[\/latex] yd, but more than the area of a rectangle with base [latex]5.6[\/latex] yd and height [latex]3.4[\/latex] yd.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 850px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224012\/CNX_BMath_Figure_09_04_066.png\" alt=\"A table with two rows. the first row is split into three columns. The first column is the formula Area of a rectangle (shown in red) equals base times height. On the next line under this it has numbers plugged into the formula; the base, 8.2 in parentheses times the height 3.4 in parentheses. Under this is it has the result 27.88 yards squared. The second column is the formula Area of a trapezoid with numbers already plugged in; one half times 3.4 yards times the quantity of 5.6 plus 8.2. Under this is has the result 23.46 yards squared. The third column is the formula Area of a rectangle (shown in blue) equals base times height. On the next line under it has number plugged into the formula; the base, 5.6 in parentheses times the height 3.4 in parentheses. Under this it has the result 19.04 yards squared. The second row shows that the Area of the red rectangle is greater than the Area of a trapezoid is greater than the Area of the blue rectangle. Beneath this, it shows the areas 27.88 for the red rectangle, 23.46 for the trapezoid, and 19.04 for the blue rectangle.\" width=\"850\" height=\"200\" \/><figcaption class=\"wp-caption-text\">Comparison of areas of rectangles and a trapezoid with the same height<\/figcaption><\/figure>\n<\/div>\n<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 409.183px;\">Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td style=\"width: 424.817px;\">Vinny has [latex]23.46[\/latex] square yards in which he can plant.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm146944\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146944&theme=lumen&iframe_resize_id=ohm146944&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<h2>Find the Area of Irregular Figures<\/h2>\n<p>So far, we have found area for rectangles, triangles, and trapezoids. An irregular figure is a figure that is not a standard geometric shape. Its area cannot be calculated using any of the standard area formulas. But some irregular figures are made up of two or more standard geometric shapes. To find the area of one of these irregular figures, we can split it into figures whose formulas we know and then add the areas of the figures.<\/p>\n<section class=\"textbox example\">Find the area of the shaded region.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 190px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224039\/CNX_BMath_Figure_09_05_012_img.png\" alt=\"An image of an attached horizontal rectangle and a vertical rectangle is shown. The top is labeled 12, the side of the horizontal rectangle is labeled 4. The side is labeled 10, the width of the vertical rectangle is labeled 2.\" width=\"190\" height=\"169\" \/><figcaption class=\"wp-caption-text\">Shaded region with sides labeled<\/figcaption><\/figure>\n<\/div>\n<div class=\"wp-nocaption \"><\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q247910\">Show Solution<\/button><\/p>\n<div id=\"q247910\" class=\"hidden-answer\" style=\"display: none\">\nThe given figure is irregular, but we can break it into two rectangles. The area of the shaded region will be the sum of the areas of both rectangles.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 222px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224040\/CNX_BMath_Figure_09_05_013_img.png\" alt=\"An image of an attached horizontal rectangle and a vertical rectangle is shown. The top is labeled 12, the side of the horizontal rectangle is labeled 4. The side is labeled 10, the width of the vertical rectangle is labeled 2.\" width=\"222\" height=\"192\" \/><figcaption class=\"wp-caption-text\">Shaded region split into two rectangles<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The blue rectangle has a width of [latex]12[\/latex] and a length of [latex]4[\/latex]. The red rectangle has a width of [latex]2[\/latex], but its length is not labeled. The right side of the figure is the length of the red rectangle plus the length of the blue rectangle. Since the right side of the blue rectangle is [latex]4[\/latex] units long, the length of the red rectangle must be [latex]6[\/latex] units.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 215px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224041\/CNX_BMath_Figure_09_05_014_img.png\" alt=\"An image of a blue horizontal rectangle attached to a red vertical rectangle is shown. The top is labeled 12, the side of the blue rectangle is labeled 4. The whole side is labeled 10, the blue portion is labeled 4 and the red portion is labeled 6. The width of the red rectangle is labeled 2.\" width=\"215\" height=\"168\" \/><figcaption class=\"wp-caption-text\">Shaded region split into two rectangles<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div style=\"text-align: center;\">\n<figure style=\"width: 205px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224043\/CNX_BMath_Figure_09_05_015_img.png\" alt=\"The first line says A sub figure equals A sub rectangle plus A sub red rectangle. Below this is A sub figure equals bh plus red bh. Below this is A sub figure equals 12 times 4 plus red 2 times 6. Below this is A sub figure equals 48 plus red 12. Below this is A sub figure equals 60.\" width=\"205\" height=\"130\" \/><figcaption class=\"wp-caption-text\">Formula for Area<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>The area of the figure is [latex]60[\/latex] square units.<\/p>\n<p>Is there another way to split this figure into two rectangles? Try it, and make sure you get the same area.\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<section>\n<section class=\"textbox example\">\n<p>Find the area of the shaded region.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 155px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224045\/CNX_BMath_Figure_09_05_018_img.png\" alt=\"A blue geometric shape is shown. It looks like a rectangle with a triangle attached to the top on the right side. The left side is labeled 4, the top 5, the bottom 8, the right side 7.\" width=\"155\" height=\"131\" \/><figcaption class=\"wp-caption-text\">Shaded region with sides labeled<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p><div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q937874\">Show Solution<\/button><\/p>\n<div id=\"q937874\" class=\"hidden-answer\" style=\"display: none\">\nWe can break this irregular figure into a triangle and rectangle. The area of the figure will be the sum of the areas of the triangle and the rectangle. The rectangle has a length of [latex]8[\/latex] units and a width of [latex]4[\/latex] units. We need to find the base and height of the triangle.<br \/>\nSince both sides of the rectangle are [latex]4[\/latex], the vertical side of the triangle is [latex]3[\/latex] , which is [latex]7 - 4[\/latex] .<br \/>\nThe length of the rectangle is [latex]8[\/latex], so the base of the triangle will be [latex]3[\/latex] , which is [latex]8 - 5[\/latex] .<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 178px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224046\/CNX_BMath_Figure_09_05_019_img.png\" alt=\"A geometric shape is shown. It is a blue rectangle with a red triangle attached to the top on the right side. The left side is labeled 4, the top 5, the bottom 8, the right side 7. The right side of the rectangle is labeled 4. The right side and bottom of the triangle are labeled 3.\" width=\"178\" height=\"131\" \/><figcaption class=\"wp-caption-text\">Shaded region split into a triangle and a rectangle<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>\nNow we can add the areas to find the area of the irregular figure.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 196px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224048\/CNX_BMath_Figure_09_05_020_img.png\" alt=\"The top line reads A sub figure equals A sub rectangle plus A sub red triangle. The second line reads A sub figure equals lw plus one-half red bh. The next line says A sub figure equals 8 times 4 plus one-half times red 3 times red 3. The next line reads A sub figure equals 32 plus red 4.5. The last line says A sub figure equals 36.5 sq. units.\" width=\"196\" height=\"169\" \/><figcaption class=\"wp-caption-text\">Formula for Area<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>\nThe area of the figure is [latex]36.5[\/latex] square units.<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\">\n<iframe loading=\"lazy\" id=\"ohm246488\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=246488&theme=lumen&iframe_resize_id=ohm246488&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><br \/>\n<\/section>\n<\/section>\n","protected":false},"author":15,"menu_order":2,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":354,"module-header":"background_you_need","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1716"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":18,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1716\/revisions"}],"predecessor-version":[{"id":4785,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1716\/revisions\/4785"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/parts\/354"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapters\/1716\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/media?parent=1716"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/pressbooks\/v2\/chapter-type?post=1716"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/contributor?post=1716"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus1\/wp-json\/wp\/v2\/license?post=1716"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}