{"id":2929,"date":"2023-05-16T20:25:24","date_gmt":"2023-05-16T20:25:24","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=2929"},"modified":"2025-08-26T03:48:37","modified_gmt":"2025-08-26T03:48:37","slug":"area-and-circumference-learn-it-4","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/area-and-circumference-learn-it-4\/","title":{"raw":"Area and Circumference: Learn It 4","rendered":"Area and Circumference: Learn It 4"},"content":{"raw":"<h2>Find the Circumference and Area of Circles<\/h2>\r\n<p>The properties of circles have been studied for over [latex]2,000[\/latex] years. All circles have exactly the same shape, but their sizes are affected by the length of the <strong>radius<\/strong>, a line segment from the center to any point on the circle. A line segment that passes through a circle\u2019s center connecting two points on the circle is called a <strong>diameter<\/strong>. The diameter is twice as long as the radius. The distance around a circle is called its <strong>circumference<\/strong>.<\/p>\r\n<center>\r\n[caption id=\"\" align=\"aligncenter\" width=\"212\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221636\/CNX_BMath_Figure_05_03_008.png\" alt=\"A circle is shown. A dotted line running through the widest portion of the circle is labeled as a diameter. A dotted line from the center of the circle to a point on the circle is labeled as a radius. Along the edge of the circle is the circumference.\" width=\"212\" height=\"212\" \/> Figure 1. The circumference, radius, and diameter of the circle are labeled[\/caption]\r\n<\/center>\r\n<p>&nbsp;<\/p>\r\n<p>Archimedes discovered that for circles of all different sizes, dividing the circumference by the diameter always gives the same number. The value of this number is pi, symbolized by Greek letter [latex]\\pi [\/latex] (pronounced \"pie\"). We approximate [latex]\\pi [\/latex] with [latex]3.14[\/latex] or [latex]\\Large\\frac{22}{7}[\/latex] depending on whether the radius of the circle is given as a decimal or a fraction.<\/p>\r\n<section class=\"textbox proTip\">If you use the [latex]\\pi [\/latex] key on your calculator to do the calculations in this section, your answers will be slightly different from the answers shown. That is because the [latex]\\pi [\/latex] key uses more than two decimal places.<\/section>\r\n<section class=\"textbox keyTakeaway\">\r\n<div>\r\n<h3>properties of circles<\/h3>\r\n<center><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224027\/CNX_BMath_Figure_09_05_001.png\" alt=\"An image of a circle is shown. There is a line drawn through the widest part at the center of the circle with a red dot indicating the center of the circle. The line is labeled d. The two segments from the center of the circle to the outside of the circle are each labeled r.\" \/><\/center>\r\n<p>&nbsp;<\/p>\r\n<br \/>\r\n<ul id=\"fs-id1489165\">\r\n\t<li>[latex]r[\/latex] is the length of the <strong>radius<\/strong><\/li>\r\n\t<li>[latex]d[\/latex] is the length of the <strong>diameter<\/strong><\/li>\r\n\t<li>[latex]d=2r[\/latex]<\/li>\r\n\t<li><strong>Circumference <\/strong>is the perimeter of a circle. The formula for circumference is [latex]C=2\\pi r[\/latex]<\/li>\r\n\t<li>The formula for <strong>area <\/strong>of a circle is [latex]A=\\pi {r}^{2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/section>\r\n<p>Since the diameter is twice the radius, another way to find the circumference is to use the formula [latex]C=\\pi \\mathit{\\text{d}}[\/latex]. Suppose we want to find the exact area of a circle of radius [latex]10[\/latex] inches. To calculate the area, we would evaluate the formula for the area when [latex]r=10[\/latex] inches and leave the answer in terms of [latex]\\pi[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ A=\\pi {\\mathit{\\text{r}}}^{2}\\hfill \\\\ A=\\pi \\text{(}{10}^{2}\\text{)}\\hfill \\\\ A=\\pi \\cdot 100\\hfill \\end{array}[\/latex]<\/p>\r\n<p>We write [latex]\\pi [\/latex] after the [latex]100[\/latex]. So the exact value of the area is [latex]A=100\\pi [\/latex] square inches. To approximate the area, we would substitute [latex]\\pi \\approx 3.14[\/latex].<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}A&amp; =&amp; 100\\pi \\hfill \\\\ \\\\ &amp; \\approx &amp; 100\\cdot 3.14\\hfill \\\\ &amp; \\approx &amp; 314\\text{ square inches}\\hfill \\end{array}[\/latex]<\/p>\r\n<p>Remember to use square units, such as square inches, when you calculate the area.<\/p>\r\n<section class=\"textbox youChoose\">[videopicker divId=\"tnh-video-picker\" title=\"Circumference and Area\" label=\"Select Video\"] [videooption displayName=\"How to Calculate the Circumference of a Circle\" value=\"\/\/plugin.3playmedia.com\/show?mf=12427424&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=sjmZwRojMWk&amp;video_target=tpm-plugin-oql1vfdl-sjmZwRojMWk\"][videooption displayName=\"Circles - Area, Circumference, Radius &amp; Diameter Explained!\" value=\"\/\/plugin.3playmedia.com\/show?mf=12427425&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=D4nGkWOPb6M&amp;video_target=tpm-plugin-wutfn29q-D4nGkWOPb6M\"] [videooption displayName=\"How to Calculate Circumference of a Circle (Step by Step) - Circumference Formula\" value=\"\/\/plugin.3playmedia.com\/show?mf=12427426&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=SzlGPN3eZcA&amp;video_target=tpm-plugin-rkod0y7r-SzlGPN3eZcA\"] [\/videopicker]\r\n\r\n<p>&nbsp;<\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+Calculate+the+Circumference+of+a+Circle.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Calculate the Circumference of a Circle\u201d here (opens in new window).<\/a><\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Circles+-+Area%2C+Circumference%2C+Radius+%26+Diameter+Explained!.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCircles - Area, Circumference, Radius &amp; Diameter Explained!\u201d here (opens in new window).<\/a><\/p>\r\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+Calculate+Circumference+of+a+Circle+(Step+by+Step)+%7C+Circumference+Formula.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Calculate Circumference of a Circle (Step by Step) | Circumference Formula\u201d here (opens in new window).<\/a><\/p>\r\n<\/section>\r\n<section class=\"textbox example\">A circular sandbox has a radius of [latex]2.5[\/latex] feet. Find the:\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Circumference of the sandbox<\/li>\r\n\t<li>Area of the sandbox<\/li>\r\n<\/ol>\r\n\r\n[reveal-answer q=\"247910\"]Show Solution[\/reveal-answer] [hidden-answer a=\"247910\"]\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>1. Circumference of the sandbox<\/th>\r\n<\/tr>\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><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224028\/CNX_BMath_Figure_09_05_029_img-01.png\" alt=\"A circle with radius labeled as 2.5 feet\" width=\"159\" height=\"159\" \/><\/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 circumference of the circle<\/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 <em>c<\/em> = circumference of the circle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula Substitute<\/td>\r\n<td>[latex]C=2\\pi r[\/latex] [latex]C=2\\pi \\left(2.5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]C\\approx 2\\left(3.14\\right)\\left(2.5\\right)[\/latex] [latex]C\\approx 15\\text{ft}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check.<\/strong> Does this answer make sense?<\/td>\r\n<td>Yes. If we draw a square around the circle, its sides would be [latex]5[\/latex] ft (twice the radius), so its perimeter would be [latex]20[\/latex] ft. This is slightly more than the circle's circumference, [latex]15.7[\/latex] ft. <img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224029\/CNX_BMath_Figure_09_05_029_img-02.png\" alt=\"A circle in a red square. The circle's radius is shown as 2.5 feet and the sides of the square are each labeled as 5 feet.\" width=\"206\" height=\"188\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The circumference of the sandbox is [latex]15.7[\/latex] feet.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>2. Area of the sandbox<\/th>\r\n<\/tr>\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><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224028\/CNX_BMath_Figure_09_05_029_img-01.png\" alt=\"A circle with radius labeled as 2.5 feet\" width=\"159\" height=\"159\" \/><\/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 circle<\/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 <em>A<\/em> = the area of the circle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula Substitute<\/td>\r\n<td>[latex]A=\\pi {r}^{2}[\/latex] [latex]A=\\pi{\\left(2.5\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]A\\approx \\left(3.14\\right){\\left(2.5\\right)}^{2}[\/latex] [latex]A\\approx 19.625\\text{ sq. ft}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check.<\/strong> Does this answer make sense?<\/td>\r\n<td>Yes. If we draw a square around the circle, its sides would be [latex]5[\/latex] ft, as shown in part 1. So the area of the square would be [latex]25[\/latex] sq. ft. This is slightly more than the circle's area, [latex]19.625[\/latex] sq. ft.<\/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 circle is [latex]19.625[\/latex] square feet.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7015[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">A circular table has a diameter of four feet. What is the circumference of the table? [reveal-answer q=\"808290\"]Show Solution[\/reveal-answer] [hidden-answer a=\"808290\"]\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><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224030\/CNX_BMath_Figure_09_05_032_img-01.png\" alt=\"A circular tabletop with a diameter labeled as 4 ft\" width=\"230\" height=\"227\" \/><\/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 circumference of the table<\/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 <em>c<\/em> = the circumference of the table<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula for the situation. Substitute.<\/td>\r\n<td>[latex]C=\\pi d[\/latex] [latex]C=\\pi \\left(4\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve<\/strong> the equation, using [latex]3.14[\/latex] for [latex]\\pi [\/latex].<\/td>\r\n<td>[latex]C\\approx \\left(3.14\\right)\\left(4\\right)[\/latex] [latex]C\\approx 12.56\\text{ feet}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check<\/strong> Does this answer make sense?<\/td>\r\n<td>If we put a square around the circle, its side would be [latex]4[\/latex]. The perimeter would be [latex]16[\/latex]. It makes sense that the circumference of the circle, [latex]12.56[\/latex], is a little less than [latex]16[\/latex]. <img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224032\/CNX_BMath_Figure_09_05_032_img-02.png\" alt=\"A circle in a red square. The diameter of the circle and each side of the square are labeled as 4 feet.\" width=\"294\" height=\"270\" \/><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\r\n<td>The diameter of the table is [latex]12.56[\/latex] square feet.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7016[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Find the diameter of a circle with a circumference of [latex]47.1[\/latex] centimeters. [reveal-answer q=\"282807\"]Show Solution[\/reveal-answer] [hidden-answer a=\"282807\"]\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><img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224033\/CNX_BMath_Figure_09_05_033_img-01.png\" alt=\"A circle with a line through the center labeled d. Beneath it, it reads C = 47.1 cm.\" width=\"218\" height=\"239\" \/> [latex]C=47.1[\/latex]cm<\/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 diameter of the circle<\/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]d[\/latex]\u00a0= the diameter of the circle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong><\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the formula. Substitute, using [latex]3.14[\/latex] to approximate [latex]\\pi [\/latex] .<\/td>\r\n<td>[latex]C=\\pi{d}[\/latex] [latex]47.1\\approx{3.14d}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex] \\Large\\frac{47.1}{3.14}\\normalsize\\approx \\Large\\frac{3.14d}{3.14}[\/latex] [latex]15\\approx{d}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check.<\/strong><\/td>\r\n<td>\r\n<p>[latex]C=\\pi{d}[\/latex]<\/p>\r\n<p>[latex]47.1\\stackrel{?}{=}\\left(3.14\\right)\\left(15\\right)[\/latex]<\/p>\r\n<p>[latex]47.1=47.1\\quad\\checkmark [\/latex]<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 7. <strong>Answer the question.<\/strong><\/td>\r\n<td>The diameter of the circle is approximately [latex]15[\/latex] centimeters.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7017[\/ohm2_question]<\/section>\r\n<h3>Area and Circumference of a Circle when Given Fractions<\/h3>\r\n<p>Sometimes we are given the diameter or radius of a circle in fractions. Recall earlier when we were given the approximations of pi, we were told [latex]\\Large\\frac{22}{7}[\/latex] is the fraction approximation of pi. If you use your calculator, the decimal number will fill up the display and show [latex]3.14285714[\/latex]. But if we round that number to two decimal places, we get [latex]3.14[\/latex], the decimal approximation of [latex]\\pi [\/latex]. When we have a circle with radius given as a fraction, we can substitute [latex]{\\Large\\frac{22}{7}}[\/latex] for [latex]\\pi [\/latex] instead of [latex]3.14[\/latex].<\/p>\r\n<section class=\"textbox example\">A circle has radius [latex]{\\Large\\frac{14}{15}}[\/latex] meters. Approximate its:\r\n\r\n<ol style=\"list-style-type: decimal;\">\r\n\t<li>Circumference<\/li>\r\n\t<li>Area<\/li>\r\n<\/ol>\r\n\r\n[reveal-answer q=\"418144\"]Show Solution[\/reveal-answer] [hidden-answer a=\"418144\"]\r\n\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>1. Circumference<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>Find the circumference when [latex]r={\\Large\\frac{14}{15}}[\/latex]<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the formula for circumference.<\/td>\r\n<td>[latex]C=2\\pi \\mathit{\\text{r}}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\Large\\frac{22}{7}[\/latex] for [latex]\\pi [\/latex] and [latex]\\Large\\frac{14}{15}[\/latex] for [latex]r[\/latex] .<\/td>\r\n<td>[latex]C\\approx 2\\left({\\Large\\frac{22}{7}}\\right)\\left({\\Large\\frac{14}{15}}\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]C\\approx {\\Large\\frac{88}{15}}\\text{meters}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table>\r\n<tbody>\r\n<tr>\r\n<th>2. Area<\/th>\r\n<\/tr>\r\n<tr>\r\n<td>Find the area when [latex]r={\\Large\\frac{14}{15}}[\/latex].<\/td>\r\n<td>&nbsp;<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the formula for area.<\/td>\r\n<td>[latex]A=\\pi {\\mathit{\\text{r}}}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Substitute [latex]\\Large\\frac{22}{7}[\/latex] for [latex]\\pi [\/latex] and [latex]\\Large\\frac{14}{15}[\/latex] for [latex]r[\/latex] .<\/td>\r\n<td>[latex]A\\approx {\\Large(\\frac{22}{7})}{\\Large(\\frac{14}{15})}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]A\\approx {\\Large\\frac{616}{225}}\\text{square meters}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[\/hidden-answer]<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]7018[\/ohm2_question]<\/section>","rendered":"<h2>Find the Circumference and Area of Circles<\/h2>\n<p>The properties of circles have been studied for over [latex]2,000[\/latex] years. All circles have exactly the same shape, but their sizes are affected by the length of the <strong>radius<\/strong>, a line segment from the center to any point on the circle. A line segment that passes through a circle\u2019s center connecting two points on the circle is called a <strong>diameter<\/strong>. The diameter is twice as long as the radius. The distance around a circle is called its <strong>circumference<\/strong>.<\/p>\n<div style=\"text-align: center;\">\n<figure style=\"width: 212px\" 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\/24221636\/CNX_BMath_Figure_05_03_008.png\" alt=\"A circle is shown. A dotted line running through the widest portion of the circle is labeled as a diameter. A dotted line from the center of the circle to a point on the circle is labeled as a radius. Along the edge of the circle is the circumference.\" width=\"212\" height=\"212\" \/><figcaption class=\"wp-caption-text\">Figure 1. The circumference, radius, and diameter of the circle are labeled<\/figcaption><\/figure>\n<\/div>\n<p>&nbsp;<\/p>\n<p>Archimedes discovered that for circles of all different sizes, dividing the circumference by the diameter always gives the same number. The value of this number is pi, symbolized by Greek letter [latex]\\pi[\/latex] (pronounced &#8220;pie&#8221;). We approximate [latex]\\pi[\/latex] with [latex]3.14[\/latex] or [latex]\\Large\\frac{22}{7}[\/latex] depending on whether the radius of the circle is given as a decimal or a fraction.<\/p>\n<section class=\"textbox proTip\">If you use the [latex]\\pi[\/latex] key on your calculator to do the calculations in this section, your answers will be slightly different from the answers shown. That is because the [latex]\\pi[\/latex] key uses more than two decimal places.<\/section>\n<section class=\"textbox keyTakeaway\">\n<div>\n<h3>properties of circles<\/h3>\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\/24224027\/CNX_BMath_Figure_09_05_001.png\" alt=\"An image of a circle is shown. There is a line drawn through the widest part at the center of the circle with a red dot indicating the center of the circle. The line is labeled d. The two segments from the center of the circle to the outside of the circle are each labeled r.\" \/><\/div>\n<p>&nbsp;<\/p>\n<div class=\"wp-nocaption \"><\/div>\n<ul id=\"fs-id1489165\">\n<li>[latex]r[\/latex] is the length of the <strong>radius<\/strong><\/li>\n<li>[latex]d[\/latex] is the length of the <strong>diameter<\/strong><\/li>\n<li>[latex]d=2r[\/latex]<\/li>\n<li><strong>Circumference <\/strong>is the perimeter of a circle. The formula for circumference is [latex]C=2\\pi r[\/latex]<\/li>\n<li>The formula for <strong>area <\/strong>of a circle is [latex]A=\\pi {r}^{2}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/section>\n<p>Since the diameter is twice the radius, another way to find the circumference is to use the formula [latex]C=\\pi \\mathit{\\text{d}}[\/latex]. Suppose we want to find the exact area of a circle of radius [latex]10[\/latex] inches. To calculate the area, we would evaluate the formula for the area when [latex]r=10[\/latex] inches and leave the answer in terms of [latex]\\pi[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{}\\\\ A=\\pi {\\mathit{\\text{r}}}^{2}\\hfill \\\\ A=\\pi \\text{(}{10}^{2}\\text{)}\\hfill \\\\ A=\\pi \\cdot 100\\hfill \\end{array}[\/latex]<\/p>\n<p>We write [latex]\\pi[\/latex] after the [latex]100[\/latex]. So the exact value of the area is [latex]A=100\\pi[\/latex] square inches. To approximate the area, we would substitute [latex]\\pi \\approx 3.14[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc}A& =& 100\\pi \\hfill \\\\ \\\\ & \\approx & 100\\cdot 3.14\\hfill \\\\ & \\approx & 314\\text{ square inches}\\hfill \\end{array}[\/latex]<\/p>\n<p>Remember to use square units, such as square inches, when you calculate the area.<\/p>\n<section class=\"textbox youChoose\">\n<div id=\"tnh-video-picker\" class=\"videoPicker\">\n<h3>Circumference and Area<\/h3>\n<form><label>Select Video:<\/label><select name=\"video\"><option value=\"\/\/plugin.3playmedia.com\/show?mf=12427424&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=sjmZwRojMWk&amp;video_target=tpm-plugin-oql1vfdl-sjmZwRojMWk\">How to Calculate the Circumference of a Circle<\/option><option value=\"\/\/plugin.3playmedia.com\/show?mf=12427425&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=D4nGkWOPb6M&amp;video_target=tpm-plugin-wutfn29q-D4nGkWOPb6M\">Circles &#8211; Area, Circumference, Radius &amp; Diameter Explained!<\/option><option value=\"\/\/plugin.3playmedia.com\/show?mf=12427426&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=SzlGPN3eZcA&amp;video_target=tpm-plugin-rkod0y7r-SzlGPN3eZcA\">How to Calculate Circumference of a Circle (Step by Step) &#8211; Circumference Formula<\/option><\/select><\/form>\n<div class=\"videoContainer threePlay\"><iframe src=\"\/\/plugin.3playmedia.com\/show?mf=12427424&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=sjmZwRojMWk&amp;video_target=tpm-plugin-oql1vfdl-sjmZwRojMWk\" allowfullscreen><\/iframe><\/div>\n<p>&nbsp;<\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+Calculate+the+Circumference+of+a+Circle.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Calculate the Circumference of a Circle\u201d here (opens in new window).<\/a><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/Circles+-+Area%2C+Circumference%2C+Radius+%26+Diameter+Explained!.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cCircles &#8211; Area, Circumference, Radius &amp; Diameter Explained!\u201d here (opens in new window).<\/a><\/p>\n<p>You can view the <a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/Quantitative+Reasoning+-+2023+Build\/Transcriptions\/How+to+Calculate+Circumference+of+a+Circle+(Step+by+Step)+%7C+Circumference+Formula.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cHow to Calculate Circumference of a Circle (Step by Step) | Circumference Formula\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">A circular sandbox has a radius of [latex]2.5[\/latex] feet. Find the:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Circumference of the sandbox<\/li>\n<li>Area of the sandbox<\/li>\n<\/ol>\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\">\n<table>\n<tbody>\n<tr>\n<th>1. Circumference of the sandbox<\/th>\n<\/tr>\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\/24224028\/CNX_BMath_Figure_09_05_029_img-01.png\" alt=\"A circle with radius labeled as 2.5 feet\" width=\"159\" height=\"159\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the circumference of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>c<\/em> = circumference of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula Substitute<\/td>\n<td>[latex]C=2\\pi r[\/latex] [latex]C=2\\pi \\left(2.5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]C\\approx 2\\left(3.14\\right)\\left(2.5\\right)[\/latex] [latex]C\\approx 15\\text{ft}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong> Does this answer make sense?<\/td>\n<td>Yes. If we draw a square around the circle, its sides would be [latex]5[\/latex] ft (twice the radius), so its perimeter would be [latex]20[\/latex] ft. This is slightly more than the circle&#8217;s circumference, [latex]15.7[\/latex] ft. <img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224029\/CNX_BMath_Figure_09_05_029_img-02.png\" alt=\"A circle in a red square. The circle's radius is shown as 2.5 feet and the sides of the square are each labeled as 5 feet.\" width=\"206\" height=\"188\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The circumference of the sandbox is [latex]15.7[\/latex] feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<th>2. Area of the sandbox<\/th>\n<\/tr>\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\/24224028\/CNX_BMath_Figure_09_05_029_img-01.png\" alt=\"A circle with radius labeled as 2.5 feet\" width=\"159\" height=\"159\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the area of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>A<\/em> = the area of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula Substitute<\/td>\n<td>[latex]A=\\pi {r}^{2}[\/latex] [latex]A=\\pi{\\left(2.5\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A\\approx \\left(3.14\\right){\\left(2.5\\right)}^{2}[\/latex] [latex]A\\approx 19.625\\text{ sq. ft}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong> Does this answer make sense?<\/td>\n<td>Yes. If we draw a square around the circle, its sides would be [latex]5[\/latex] ft, as shown in part 1. So the area of the square would be [latex]25[\/latex] sq. ft. This is slightly more than the circle&#8217;s area, [latex]19.625[\/latex] sq. ft.<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The area of the circle is [latex]19.625[\/latex] square feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7015\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7015&theme=lumen&iframe_resize_id=ohm7015&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">A circular table has a diameter of four feet. What is the circumference of the table? <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q808290\">Show Solution<\/button> <\/p>\n<div id=\"q808290\" 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\/24224030\/CNX_BMath_Figure_09_05_032_img-01.png\" alt=\"A circular tabletop with a diameter labeled as 4 ft\" width=\"230\" height=\"227\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the circumference of the table<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let <em>c<\/em> = the circumference of the table<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong> Write the appropriate formula for the situation. Substitute.<\/td>\n<td>[latex]C=\\pi d[\/latex] [latex]C=\\pi \\left(4\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve<\/strong> the equation, using [latex]3.14[\/latex] for [latex]\\pi[\/latex].<\/td>\n<td>[latex]C\\approx \\left(3.14\\right)\\left(4\\right)[\/latex] [latex]C\\approx 12.56\\text{ feet}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check<\/strong> Does this answer make sense?<\/td>\n<td>If we put a square around the circle, its side would be [latex]4[\/latex]. The perimeter would be [latex]16[\/latex]. It makes sense that the circumference of the circle, [latex]12.56[\/latex], is a little less than [latex]16[\/latex]. <img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24224032\/CNX_BMath_Figure_09_05_032_img-02.png\" alt=\"A circle in a red square. The diameter of the circle and each side of the square are labeled as 4 feet.\" width=\"294\" height=\"270\" \/><\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer<\/strong> the question.<\/td>\n<td>The diameter of the table is [latex]12.56[\/latex] square feet.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7016\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7016&theme=lumen&iframe_resize_id=ohm7016&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Find the diameter of a circle with a circumference of [latex]47.1[\/latex] centimeters. <\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q282807\">Show Solution<\/button> <\/p>\n<div id=\"q282807\" 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\/24224033\/CNX_BMath_Figure_09_05_033_img-01.png\" alt=\"A circle with a line through the center labeled d. Beneath it, it reads C = 47.1 cm.\" width=\"218\" height=\"239\" \/> [latex]C=47.1[\/latex]cm<\/td>\n<\/tr>\n<tr>\n<td>Step 2. <strong>Identify<\/strong> what you are looking for.<\/td>\n<td>the diameter of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 3. <strong>Name.<\/strong> Choose a variable to represent it.<\/td>\n<td>Let [latex]d[\/latex]\u00a0= the diameter of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Write the formula. Substitute, using [latex]3.14[\/latex] to approximate [latex]\\pi[\/latex] .<\/td>\n<td>[latex]C=\\pi{d}[\/latex] [latex]47.1\\approx{3.14d}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]\\Large\\frac{47.1}{3.14}\\normalsize\\approx \\Large\\frac{3.14d}{3.14}[\/latex] [latex]15\\approx{d}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/td>\n<td>\n[latex]C=\\pi{d}[\/latex]<br \/>\n[latex]47.1\\stackrel{?}{=}\\left(3.14\\right)\\left(15\\right)[\/latex]<br \/>\n[latex]47.1=47.1\\quad\\checkmark[\/latex]\n<\/td>\n<\/tr>\n<tr>\n<td>Step 7. <strong>Answer the question.<\/strong><\/td>\n<td>The diameter of the circle is approximately [latex]15[\/latex] centimeters.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7017\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7017&theme=lumen&iframe_resize_id=ohm7017&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<h3>Area and Circumference of a Circle when Given Fractions<\/h3>\n<p>Sometimes we are given the diameter or radius of a circle in fractions. Recall earlier when we were given the approximations of pi, we were told [latex]\\Large\\frac{22}{7}[\/latex] is the fraction approximation of pi. If you use your calculator, the decimal number will fill up the display and show [latex]3.14285714[\/latex]. But if we round that number to two decimal places, we get [latex]3.14[\/latex], the decimal approximation of [latex]\\pi[\/latex]. When we have a circle with radius given as a fraction, we can substitute [latex]{\\Large\\frac{22}{7}}[\/latex] for [latex]\\pi[\/latex] instead of [latex]3.14[\/latex].<\/p>\n<section class=\"textbox example\">A circle has radius [latex]{\\Large\\frac{14}{15}}[\/latex] meters. Approximate its:<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>Circumference<\/li>\n<li>Area<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q418144\">Show Solution<\/button> <\/p>\n<div id=\"q418144\" class=\"hidden-answer\" style=\"display: none\">\n<table>\n<tbody>\n<tr>\n<th>1. Circumference<\/th>\n<\/tr>\n<tr>\n<td>Find the circumference when [latex]r={\\Large\\frac{14}{15}}[\/latex]<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Write the formula for circumference.<\/td>\n<td>[latex]C=2\\pi \\mathit{\\text{r}}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\Large\\frac{22}{7}[\/latex] for [latex]\\pi[\/latex] and [latex]\\Large\\frac{14}{15}[\/latex] for [latex]r[\/latex] .<\/td>\n<td>[latex]C\\approx 2\\left({\\Large\\frac{22}{7}}\\right)\\left({\\Large\\frac{14}{15}}\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]C\\approx {\\Large\\frac{88}{15}}\\text{meters}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table>\n<tbody>\n<tr>\n<th>2. Area<\/th>\n<\/tr>\n<tr>\n<td>Find the area when [latex]r={\\Large\\frac{14}{15}}[\/latex].<\/td>\n<td>&nbsp;<\/td>\n<\/tr>\n<tr>\n<td>Write the formula for area.<\/td>\n<td>[latex]A=\\pi {\\mathit{\\text{r}}}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Substitute [latex]\\Large\\frac{22}{7}[\/latex] for [latex]\\pi[\/latex] and [latex]\\Large\\frac{14}{15}[\/latex] for [latex]r[\/latex] .<\/td>\n<td>[latex]A\\approx {\\Large(\\frac{22}{7})}{\\Large(\\frac{14}{15})}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]A\\approx {\\Large\\frac{616}{225}}\\text{square meters}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm7018\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=7018&theme=lumen&iframe_resize_id=ohm7018&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":15,"menu_order":21,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"How to Calculate the Circumference of a Circle\",\"author\":\"wikiHow\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/sjmZwRojMWk\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"Circles - Area, Circumference, Radius & Diameter Explained!\",\"author\":\"The Organic Chemistry Tutor\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/D4nGkWOPb6M\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"},{\"type\":\"copyrighted_video\",\"description\":\"How to Calculate Circumference of a Circle (Step by Step) | Circumference Formula\",\"author\":\"Math with Mr. J\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/SzlGPN3eZcA\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"\"}]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":71,"module-header":"learn_it","content_attributions":[{"type":"copyrighted_video","description":"How to Calculate the Circumference of a Circle","author":"wikiHow","organization":"","url":"https:\/\/youtu.be\/sjmZwRojMWk","project":"","license":"arr","license_terms":""},{"type":"copyrighted_video","description":"Circles - Area, Circumference, Radius & Diameter Explained!","author":"The Organic Chemistry Tutor","organization":"","url":"https:\/\/youtu.be\/D4nGkWOPb6M","project":"","license":"arr","license_terms":""},{"type":"copyrighted_video","description":"How to Calculate Circumference of a Circle (Step by Step) | Circumference Formula","author":"Math with Mr. J","organization":"","url":"https:\/\/youtu.be\/SzlGPN3eZcA","project":"","license":"arr","license_terms":""}],"internal_book_links":[],"video_content":[{"divId":"tnh-video-picker","title":"Circumference and Area","label":"Select Video","video_collection":[{"displayName":"How to Calculate the Circumference of a Circle","value":"\/\/plugin.3playmedia.com\/show?mf=12427424&amp;p3sdk_version=1.10.1&amp;p=20361&amp;pt=375&amp;video_id=sjmZwRojMWk&amp;video_target=tpm-plugin-oql1vfdl-sjmZwRojMWk"},{"displayName":"Circles - Area, Circumference, Radius &amp; 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