{"id":10772,"date":"2017-06-05T16:15:11","date_gmt":"2017-06-05T16:15:11","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/prealgebra\/?post_type=chapter&#038;p=10772"},"modified":"2018-01-13T19:17:47","modified_gmt":"2018-01-13T19:17:47","slug":"using-the-properties-of-circles-to-solve-problems","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-prealgebra\/chapter\/using-the-properties-of-circles-to-solve-problems\/","title":{"raw":"Using the Properties of Circles to Solve Problems","rendered":"Using the Properties of Circles to Solve Problems"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the circumference and area of a circular object given it's radius<\/li>\r\n \t<li>Find the circumference and area of a circular object given it's diameter<\/li>\r\n \t<li>Calculate the diameter of a circular object given it's circumference<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p data-type=\"title\">Do you remember the properties of circles from Decimals and Fractions Together? We\u2019ll show them here again to refer to as we use them to solve applications.<\/p>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Properties of Circles<\/h3>\r\n<img 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.\" data-media-type=\"image\/png\" \/>\r\n<ul id=\"fs-id1489165\" data-bullet-style=\"bullet\">\r\n \t<li>[latex]r[\/latex] is the length of the radius<\/li>\r\n \t<li>[latex]d[\/latex] is the length of the diameter<\/li>\r\n \t<li>[latex]d=2r[\/latex]<\/li>\r\n \t<li>Circumference 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 area of a circle is[latex]A=\\pi {r}^{2}[\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\nRemember, that 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. 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.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA circular sandbox has a radius of [latex]2.5[\/latex] feet. Find the 1. circumference and 2. area of the sandbox.\r\n\r\nSolution\r\n<table id=\"eip-id1168467555346\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>1.\r\n\r\nStep 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><img 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=\".\" data-media-type=\"image\/png\" \/><\/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 data-effect=\"italics\">c<\/em> = circumference of the circle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula\r\n\r\nSubstitute<\/td>\r\n<td>[latex]C=2\\pi r[\/latex]\r\n\r\n[latex]C=2\\pi \\left(2.5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]C\\approx 2\\left(3.14\\right)\\left(2.5\\right)[\/latex]\r\n\r\n[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?\r\n\r\nYes. 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.\r\n\r\n<img 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=\".\" data-media-type=\"image\/png\" \/><\/td>\r\n<td><\/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 id=\"eip-id1168468253524\" class=\"unnumbered unstyled\" summary=\"Identify what you are looking for.\" data-label=\"\">\r\n<tbody>\r\n<tr>\r\n<td>2.\r\n\r\nStep 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n<td><img 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=\".\" data-media-type=\"image\/png\" \/><\/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 data-effect=\"italics\">A<\/em> = the area of the circle<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula\r\n\r\nSubstitute<\/td>\r\n<td>[latex]A=\\pi {r}^{2}[\/latex]\r\n\r\n[latex]A=\\pi{\\left(2.5\\right)}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\r\n<td>[latex]A\\approx \\left(3.14\\right){\\left(2.5\\right)}^{2}[\/latex]\r\n\r\n[latex]A\\approx 19.625\\text{sq. ft}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check.<\/strong>\r\n\r\nYes. If we draw a square around the circle, its sides would be [latex]5[\/latex] ft, as shown in part \u24d0. 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<td><\/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<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146563[\/ohm_question]\r\n\r\n[ohm_question]146564[\/ohm_question]\r\n\r\n<\/div>\r\nin the following video we show another example of how to find the area of a circle.\r\n\r\nhttps:\/\/youtu.be\/SIKkWLqt2mQ\r\n\r\nWe usually see the formula for circumference in terms of the radius [latex]r[\/latex] of the circle:\r\n\r\n[latex]C=2\\pi r[\/latex]\r\n\r\nBut since the diameter of a circle is two times the radius, we could write the formula for the circumference in terms [latex]\\text{of }d[\/latex].\r\n\r\n[latex]\\begin{array}{cccc}&amp; &amp; &amp; C=2\\pi r\\hfill \\\\ \\text{Using the commutative property, we get}\\hfill &amp; &amp; &amp; C=\\pi \\cdot 2r\\hfill \\\\ \\text{Then substituting}d=2r\\hfill &amp; &amp; &amp; C=\\pi \\cdot d\\hfill \\\\ \\text{So}\\hfill &amp; &amp; &amp; C=\\pi d\\hfill \\end{array}[\/latex]\r\n\r\nWe will use this form of the circumference when we\u2019re given the length of the diameter instead of the radius.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA circular table has a diameter of four feet. What is the circumference of the table?\r\n[reveal-answer q=\"808290\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"808290\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168467247251\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \" data-label=\"\">\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 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=\".\" data-media-type=\"image\/png\" \/><\/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 data-effect=\"italics\">c<\/em> = the circumference of the table<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 4. <strong>Translate.<\/strong>\r\n\r\nWrite the appropriate formula for the situation.\r\n\r\nSubstitute.<\/td>\r\n<td>[latex]C=\\pi d[\/latex]\r\n\r\n[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]\r\n\r\n[latex]C\\approx 12.56\\text{feet}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong> If we put a square around the circle, its side would be [latex]4[\/latex].\r\n\r\nThe 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].\r\n\r\n<img 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=\".\" data-media-type=\"image\/png\" \/><\/td>\r\n<td><\/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[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146785[\/ohm_question]\r\n\r\n[ohm_question]146786[\/ohm_question]\r\n\r\n<\/div>\r\nIn the next video we show two more examples of how to find the circumference of a circle given it's diameter, or it's radius.\r\n\r\nhttps:\/\/youtu.be\/sHtsnC2Mgnk\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nFind the diameter of a circle with a circumference of [latex]47.1[\/latex] centimeters.\r\n[reveal-answer q=\"282807\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"282807\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468767882\" class=\"unnumbered unstyled\" summary=\"Step 1 says, \" data-label=\"\">\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 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=\".\" data-media-type=\"image\/png\" \/>\r\n\r\n[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><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Write the formula.\r\n\r\nSubstitute, using [latex]3.14[\/latex] to approximate [latex]\\pi [\/latex] .<\/td>\r\n<td>[latex]C=\\pi{d}[\/latex]\r\n\r\n[latex]47.1\\approx{3.14d}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-valign=\"top\">Step 5. <strong>Solve.<\/strong><\/td>\r\n<td>[latex]\r\n\r\n\\Large\\frac{47.1}{3.14}\\normalsize\\approx\r\n\r\n\\Large\\frac{3.14d}{3.14}[\/latex]\r\n\r\n[latex]15\\approx{d}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Step 6. <strong>Check:<\/strong>\r\n\r\n<strong>[latex]C=\\pi{d}[\/latex]<\/strong>\r\n\r\n[latex]47.1\\stackrel{?}{=}\\left(3.14\\right)\\left(15\\right)[\/latex]\r\n\r\n[latex]47.1=47.1\\quad\\checkmark [\/latex]<\/td>\r\n<td><\/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[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it<\/h3>\r\n[ohm_question]146787[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find the circumference and area of a circular object given it&#8217;s radius<\/li>\n<li>Find the circumference and area of a circular object given it&#8217;s diameter<\/li>\n<li>Calculate the diameter of a circular object given it&#8217;s circumference<\/li>\n<\/ul>\n<\/div>\n<p data-type=\"title\">Do you remember the properties of circles from Decimals and Fractions Together? We\u2019ll show them here again to refer to as we use them to solve applications.<\/p>\n<div class=\"textbox shaded\">\n<h3>Properties of Circles<\/h3>\n<p><img decoding=\"async\" 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.\" data-media-type=\"image\/png\" \/><\/p>\n<ul id=\"fs-id1489165\" data-bullet-style=\"bullet\">\n<li>[latex]r[\/latex] is the length of the radius<\/li>\n<li>[latex]d[\/latex] is the length of the diameter<\/li>\n<li>[latex]d=2r[\/latex]<\/li>\n<li>Circumference is the perimeter of a circle. The formula for circumference is[latex]C=2\\pi r[\/latex]<\/li>\n<li>The formula for area of a circle is[latex]A=\\pi {r}^{2}[\/latex]<\/li>\n<\/ul>\n<\/div>\n<p>Remember, that 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. 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.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>A circular sandbox has a radius of [latex]2.5[\/latex] feet. Find the 1. circumference and 2. area of the sandbox.<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168467555346\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\" data-label=\"\">\n<tbody>\n<tr>\n<td>1.<\/p>\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img decoding=\"async\" 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=\".\" data-media-type=\"image\/png\" \/><\/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 data-effect=\"italics\">c<\/em> = circumference of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula<\/p>\n<p>Substitute<\/td>\n<td>[latex]C=2\\pi r[\/latex]<\/p>\n<p>[latex]C=2\\pi \\left(2.5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]C\\approx 2\\left(3.14\\right)\\left(2.5\\right)[\/latex]<\/p>\n<p>[latex]C\\approx 15\\text{ft}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong> Does this answer make sense?<\/p>\n<p>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.<\/p>\n<p><img decoding=\"async\" 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=\".\" data-media-type=\"image\/png\" \/><\/td>\n<td><\/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 id=\"eip-id1168468253524\" class=\"unnumbered unstyled\" summary=\"Identify what you are looking for.\" data-label=\"\">\n<tbody>\n<tr>\n<td>2.<\/p>\n<p>Step 1. <strong>Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\n<td><img decoding=\"async\" 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=\".\" data-media-type=\"image\/png\" \/><\/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 data-effect=\"italics\">A<\/em> = the area of the circle<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula<\/p>\n<p>Substitute<\/td>\n<td>[latex]A=\\pi {r}^{2}[\/latex]<\/p>\n<p>[latex]A=\\pi{\\left(2.5\\right)}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 5. <strong>Solve<\/strong> the equation.<\/td>\n<td>[latex]A\\approx \\left(3.14\\right){\\left(2.5\\right)}^{2}[\/latex]<\/p>\n<p>[latex]A\\approx 19.625\\text{sq. ft}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check.<\/strong><\/p>\n<p>Yes. If we draw a square around the circle, its sides would be [latex]5[\/latex] ft, as shown in part \u24d0. 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<td><\/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<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146563\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146563&theme=oea&iframe_resize_id=ohm146563&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146564\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146564&theme=oea&iframe_resize_id=ohm146564&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>in the following video we show another example of how to find the area of a circle.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Example:  Determine the Area of a Circle\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/SIKkWLqt2mQ?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>We usually see the formula for circumference in terms of the radius [latex]r[\/latex] of the circle:<\/p>\n<p>[latex]C=2\\pi r[\/latex]<\/p>\n<p>But since the diameter of a circle is two times the radius, we could write the formula for the circumference in terms [latex]\\text{of }d[\/latex].<\/p>\n<p>[latex]\\begin{array}{cccc}& & & C=2\\pi r\\hfill \\\\ \\text{Using the commutative property, we get}\\hfill & & & C=\\pi \\cdot 2r\\hfill \\\\ \\text{Then substituting}d=2r\\hfill & & & C=\\pi \\cdot d\\hfill \\\\ \\text{So}\\hfill & & & C=\\pi d\\hfill \\end{array}[\/latex]<\/p>\n<p>We will use this form of the circumference when we\u2019re given the length of the diameter instead of the radius.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>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\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q808290\">Show Solution<\/span><\/p>\n<div id=\"q808290\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168467247251\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\" data-label=\"\">\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 decoding=\"async\" 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=\".\" data-media-type=\"image\/png\" \/><\/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 data-effect=\"italics\">c<\/em> = the circumference of the table<\/td>\n<\/tr>\n<tr>\n<td>Step 4. <strong>Translate.<\/strong><\/p>\n<p>Write the appropriate formula for the situation.<\/p>\n<p>Substitute.<\/td>\n<td>[latex]C=\\pi d[\/latex]<\/p>\n<p>[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]<\/p>\n<p>[latex]C\\approx 12.56\\text{feet}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong> If we put a square around the circle, its side would be [latex]4[\/latex].<\/p>\n<p>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].<\/p>\n<p><img decoding=\"async\" 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=\".\" data-media-type=\"image\/png\" \/><\/td>\n<td><\/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<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146785\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146785&theme=oea&iframe_resize_id=ohm146785&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<p><iframe loading=\"lazy\" id=\"ohm146786\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146786&theme=oea&iframe_resize_id=ohm146786&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<p>In the next video we show two more examples of how to find the circumference of a circle given it&#8217;s diameter, or it&#8217;s radius.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Examples:  Determine the Circumference of a Circle\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/sHtsnC2Mgnk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>Find the diameter of a circle with a circumference of [latex]47.1[\/latex] centimeters.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q282807\">Show Solution<\/span><\/p>\n<div id=\"q282807\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468767882\" class=\"unnumbered unstyled\" summary=\"Step 1 says,\" data-label=\"\">\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 decoding=\"async\" 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=\".\" data-media-type=\"image\/png\" \/><\/p>\n<p>[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><\/td>\n<\/tr>\n<tr>\n<td>Write the formula.<\/p>\n<p>Substitute, using [latex]3.14[\/latex] to approximate [latex]\\pi[\/latex] .<\/td>\n<td>[latex]C=\\pi{d}[\/latex]<\/p>\n<p>[latex]47.1\\approx{3.14d}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-valign=\"top\">Step 5. <strong>Solve.<\/strong><\/td>\n<td>[latex]\\Large\\frac{47.1}{3.14}\\normalsize\\approx    \\Large\\frac{3.14d}{3.14}[\/latex]<\/p>\n<p>[latex]15\\approx{d}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Step 6. <strong>Check:<\/strong><\/p>\n<p><strong>[latex]C=\\pi{d}[\/latex]<\/strong><\/p>\n<p>[latex]47.1\\stackrel{?}{=}\\left(3.14\\right)\\left(15\\right)[\/latex]<\/p>\n<p>[latex]47.1=47.1\\quad\\checkmark[\/latex]<\/td>\n<td><\/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<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146787\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146787&theme=oea&iframe_resize_id=ohm146787&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-10772\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Question ID 146787, 146786, 146785. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Examples: Determine the Circumference of a Circle. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/sHtsnC2Mgnk\">https:\/\/youtu.be\/sHtsnC2Mgnk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Example: Determine the Area of a Circle. <strong>Authored by<\/strong>: James Sousa (mathispower4u.com). <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/SIKkWLqt2mQ\">https:\/\/youtu.be\/SIKkWLqt2mQ<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Prealgebra. <strong>Provided by<\/strong>: OpenStax. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":17533,"menu_order":14,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Prealgebra\",\"author\":\"\",\"organization\":\"OpenStax\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"Download for free at http:\/\/cnx.org\/contents\/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757\"},{\"type\":\"original\",\"description\":\"Question ID 146787, 146786, 146785\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Examples: Determine the Circumference of a Circle\",\"author\":\"James Sousa 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