{"id":373,"date":"2025-02-13T19:44:24","date_gmt":"2025-02-13T19:44:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus2\/chapter\/introduction-to-integration-background-youll-need-1\/"},"modified":"2025-02-13T19:44:24","modified_gmt":"2025-02-13T19:44:24","slug":"introduction-to-integration-background-youll-need-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus2\/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":"\n<section class=\"textbox learningGoals\">\n<ul>\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>\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<center><img class=\"aligncenter\" 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\"><\/center>\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<p><img class=\"aligncenter\" 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\"><\/p>\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\t<li>Rectangles have four sides and four right [latex]\\left(\\text{90}^ \\circ\\right)[\/latex] angles.<\/li>\n\t<li>The lengths of opposite sides are equal.<\/li>\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>\n<\/ul>\n<center>[latex]A=L\\cdot W[\/latex]<\/center><\/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.<br>\n[reveal-answer q=\"172561\"]Show Solution[\/reveal-answer] [hidden-answer a=\"172561\"]\n\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><img class=\"alignnone\" 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\"><\/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><img class=\"alignnone\" 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\"><\/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<p>[latex]A\\stackrel{?}{=}640[\/latex]<\/p>\n<p>[latex]32\\cdot 20\\stackrel{?}{=}640[\/latex]<\/p>\n<p>[latex]640=640\\checkmark[\/latex]<\/p>\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\n[\/hidden-answer]<\/section>\n<section class=\"textbox tryIt\">[ohm_question hide_question_numbers=1]288389[\/ohm_question]<\/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<center><img class=\"aligncenter\" 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\"><\/center>\n<p>&nbsp;<\/p>\n<p><br>\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<center><img class=\"aligncenter\" 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\"><\/center>\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&nbsp;shows three triangles with the base and height of each marked.<\/p>\n<center><img class=\"aligncenter\" 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\"><\/center>\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<br>\n<center><img class=\"aligncenter\" 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\"><\/center>\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.<br>\n[reveal-answer q=\"247910\"]Show Solution[\/reveal-answer]<br>\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.<img class=\"alignnone\" 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\">Step 7. <strong>Answer<\/strong> the question.The area is [latex]44[\/latex] square inches.\n\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;\"><img class=\"alignnone\" 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\"><\/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<p>[latex]A=\\frac{1}{2}bh[\/latex]<\/p>\n<p>[latex]44\\stackrel{?}{=}\\frac{1}{2}(11)8[\/latex]<\/p>\n<p>[latex]44=44\\quad\\checkmark[\/latex]<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]288390[\/ohm_question]<\/p>\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<center><img class=\"aligncenter\" 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\"><\/center>\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<center><img class=\"aligncenter\" 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\"><\/center>\n<p>&nbsp;<\/p>\n<p>The height of the trapezoid is also the height of each of the two triangles.<\/p>\n<center><img class=\"aligncenter\" 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\"><\/center>\n<p>&nbsp;<\/p>\n<p>The formula for the area of a trapezoid is<\/p>\n<center><img class=\"aligncenter\" 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\"><\/center>\n<p>&nbsp;<\/p>\n<p>If we distribute, we get,<\/p>\n<center><img class=\"aligncenter\" 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.\"><\/center>\n<p>&nbsp;<\/p>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>properties of trapezoids<\/h3>\n<ul id=\"fs-id1429217\">\n\t<li>A <strong>trapezoid<\/strong> has four sides.<\/li>\n\t<li>Two of its sides are parallel and two sides are not.<\/li>\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>\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. [reveal-answer q=\"247911\"]Show Solution[\/reveal-answer] [hidden-answer a=\"247911\"]\n\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><img class=\"alignnone\" 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\"><\/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><img class=\"alignnone\" 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\"><\/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>&nbsp;[latex]\\checkmark[\/latex]&nbsp; see reasoning below<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n\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><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.\"><\/center>\n<p>&nbsp;<\/p>\n\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>\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? [reveal-answer q=\"676574\"]Show Solution[\/reveal-answer] [hidden-answer a=\"676574\"]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;\"><img class=\"alignnone\" 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\"><\/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;\"><img class=\"alignnone\" 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\"><\/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.<center><img class=\"alignnone\" 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.\"><\/center><\/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<p>[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]146944[\/ohm_question]<\/p>\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.<br>\n<center><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.\"><\/center><br>\n[reveal-answer q=\"247910\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"247910\"]<br>\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>\n<center><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.\"><\/center>\n<p>&nbsp;<\/p>\n\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><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.\"><\/center>\n<p>&nbsp;<\/p>\n<br>\n<center><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.\"><\/center>\n<p>&nbsp;<\/p>\n\nThe area of the figure is [latex]60[\/latex] square units.<br>\n<br>\nIs there another way to split this figure into two rectangles? Try it, and make sure you get the same area.<br>\n[\/hidden-answer]<\/section>\n<\/section>\n<section>\n<section class=\"textbox example\">\n<p>Find the area of the shaded region.<\/p>\n<center><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.\"><\/center>\n<p>&nbsp;<\/p>\n<p>[reveal-answer q=\"937874\"]Show Solution[\/reveal-answer]<br>\n[hidden-answer a=\"937874\"]<br>\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<center><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.\"><\/center>\n<p>&nbsp;<\/p>\n<p><br>\nNow we can add the areas to find the area of the irregular figure.<\/p>\n<center><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.\"><\/center>\n<p>&nbsp;<\/p>\n<p><br>\nThe area of the figure is [latex]36.5[\/latex] square units.[\/hidden-answer]<\/p>\n<\/section>\n<section class=\"textbox tryIt\">\n<p>[ohm_question hide_question_numbers=1]246488[\/ohm_question]<\/p>\n<\/section>\n<\/section>\n","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;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/p>\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><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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\" \/><\/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><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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\" \/><\/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;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/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;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/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&nbsp;shows three triangles with the base and height of each marked.<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/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;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/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.<img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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\" \/>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;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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\" \/><\/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;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/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;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/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;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>The formula for the area of a trapezoid is<\/p>\n<div style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" 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\" \/><\/div>\n<p>&nbsp;<\/p>\n<p>If we distribute, we get,<\/p>\n<div style=\"text-align: center;\"><img decoding=\"async\" class=\"aligncenter\" 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.\" \/><\/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><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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\" \/><\/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><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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\" \/><\/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>&nbsp;[latex]\\checkmark[\/latex]&nbsp; 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;\"><img 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.\" \/><\/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;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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\" \/><\/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;\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" 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\" \/><\/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;\"><img decoding=\"async\" class=\"alignnone\" 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.\" \/><\/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;\"><img 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.\" \/><\/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;\"><img 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.\" \/><\/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;\"><img 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.\" \/><\/div>\n<p>&nbsp;<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<div style=\"text-align: center;\"><img 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.\" \/><\/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;\"><img 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.\" \/><\/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;\"><img 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.\" \/><\/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;\"><img 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.\" \/><\/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":6,"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":371,"module-header":"","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\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/373"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/users\/6"}],"version-history":[{"count":0,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/373\/revisions"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/parts\/371"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapters\/373\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/media?parent=373"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/pressbooks\/v2\/chapter-type?post=373"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/contributor?post=373"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/calculus2\/wp-json\/wp\/v2\/license?post=373"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}