{"id":4436,"date":"2020-04-13T13:25:36","date_gmt":"2020-04-13T13:25:36","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=4436"},"modified":"2021-02-06T00:04:22","modified_gmt":"2021-02-06T00:04:22","slug":"using-decimals-and-fractions-with-circles","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/using-decimals-and-fractions-with-circles\/","title":{"raw":"Using Decimals and Fractions With Circles","rendered":"Using Decimals and Fractions With Circles"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find the circumference of a circle<\/li>\r\n \t<li>Find the area of a circle<\/li>\r\n<\/ul>\r\n<\/div>\r\nThe 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 radius, 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 diameter. The diameter is twice as long as the radius. See the image below.\r\n\r\nThe size of a circle can be measured in two ways. The distance around a circle is called its circumference.\r\n\r\n<img class=\"aligncenter\" 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.\" \/>\r\nArchimedes 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\"). However, the exact value of [latex]\\pi [\/latex] cannot be calculated since the decimal never ends or repeats (we will learn more about numbers like this in The Properties of Real Numbers.)\r\n\r\nDoing the Manipulative Mathematics activity Pi Lab will help you develop a better understanding of pi.\r\n\r\nIf we want the exact circumference or area of a circle, we leave the symbol [latex]\\pi [\/latex] in the answer. We can get an approximate answer by substituting [latex]3.14[\/latex] as the value of [latex]\\pi [\/latex]. We use the symbol [latex]\\approx [\/latex] to show that the result is approximate, not exact.\r\n<div class=\"textbox shaded\">\r\n<h3>Properties of Circles<\/h3>\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221637\/CNX_BMath_Figure_05_03_010_img.png\" alt=\"A circle is shown. A line runs through the widest portion of the circle. There is a red dot at the center of the circle. The half of the line from the center of the circle to a point on the right of the circle is labeled with an r. The half of the line from the center of the circle to a point on the left of the circle is also labeled with an r. The two sections labeled r have a brace drawn underneath showing that the entire segment is labeled d.\" \/>\r\n\r\n[latex]\\begin{array}{c}r\\text{ is the length of the radius.}\\hfill \\\\ d\\text{ is the length of the diameter.}\\hfill \\end{array}[\/latex]\r\n[latex]\\begin{array}{cccc}\\text{The circumference is }2\\pi \\mathit{\\text{r}}.\\hfill &amp; &amp; &amp; C=2\\pi \\mathit{\\text{r}}\\hfill \\\\ \\text{The area is }\\pi{\\mathit{\\text{r}}}^{2}.\\hfill &amp; &amp; &amp; A=\\pi{\\mathit{\\text{r}}}^{2}\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\nSince the diameter is twice the radius, another way to find the circumference is to use the formula [latex]C=\\pi \\mathit{\\text{d}}[\/latex].\r\n\r\nSuppose 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].\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 style=\"text-align: left\">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.\r\nTo 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\nRemember to use square units, such as square inches, when you calculate the area.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA circle has radius [latex]10[\/latex] centimeters.Approximate its circumference and area.\r\n\r\nSolution\r\n<table id=\"eip-id1168467143703\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.\u00a0 Find the circumference when [latex]r=10[\/latex].<\/td>\r\n<td><\/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]3.14[\/latex] for [latex]\\pi [\/latex] and 10 for , [latex]r[\/latex] .<\/td>\r\n<td>[latex]C\\approx 2\\left(3.14\\right)\\left(10\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]C\\approx 62.8\\text{ centimeters}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168469856534\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.\u00a0 Find the area when [latex]r=10[\/latex].<\/td>\r\n<td><\/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]3.14[\/latex] for [latex]\\pi [\/latex] and 10 for [latex]r[\/latex] .<\/td>\r\n<td>[latex]A\\approx \\left(3.14\\right){\\text{(}10\\text{)}}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]A\\approx 314\\text{ square centimeters}[\/latex]<\/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<\/div>\r\n&nbsp;\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA circle has radius [latex]42.5[\/latex] centimeters. Approximate its circumference and area.\r\n[reveal-answer q=\"272573\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"272573\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168468436463\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.\u00a0 Find the circumference when [latex]r=42.5[\/latex]<\/td>\r\n<td><\/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]3.14[\/latex] for [latex]\\pi [\/latex] and [latex]42.5[\/latex] for [latex]r[\/latex]<\/td>\r\n<td>[latex]C\\approx 2\\left(3.14\\right)\\left(42.5\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Multiply.<\/td>\r\n<td>[latex]C\\approx 266.9\\text{ centimeters}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<table id=\"eip-id1168466204784\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">2.\u00a0 Find the area when [latex]r=42.5[\/latex]<\/td>\r\n<td style=\"height: 15px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px\">\r\n<td style=\"height: 15px\">Write the formula for area.<\/td>\r\n<td style=\"height: 15px\">[latex]A=\\pi {\\mathit{\\text{r}}}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 31px\">\r\n<td style=\"height: 31px\">Substitute [latex]3.14[\/latex] for [latex]\\pi [\/latex] and [latex]42.5[\/latex] for [latex]r[\/latex] .<\/td>\r\n<td style=\"height: 31px\">[latex]A\\approx \\left(3.14\\right){\\text{(}42.5\\text{)}}^{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 31px\">\r\n<td style=\"height: 31px\">Multiply.<\/td>\r\n<td style=\"height: 31px\">[latex]A\\approx 5671.625\\text{ square centimeters}[\/latex]<\/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]146564[\/ohm_question]\r\n\r\n<\/div>\r\nWatch the following video to see another example of how to find the circumference of a circle.\r\n\r\nhttps:\/\/youtu.be\/sHtsnC2Mgnk\r\n\r\nIn the next video example, we find the area of a circle.\r\n\r\nhttps:\/\/youtu.be\/SIKkWLqt2mQ\r\n<h2>Approximate [latex]\\pi [\/latex] with a Fraction<\/h2>\r\nConvert the fraction [latex]{\\Large\\frac{22}{7}}[\/latex] to a decimal. 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]. And, since [latex]{\\Large\\frac{22}{7}}[\/latex] is also an approximation of [latex]\\pi [\/latex], we will use the [latex]\\approx [\/latex] symbol to show we have an approximate value.\r\n<div class=\"textbox exercises\">\r\n<h3>example<\/h3>\r\nA circle has radius [latex]{\\Large\\frac{14}{15}}[\/latex] meters. Approximate its circumference and area.\r\n[reveal-answer q=\"418144\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"418144\"]\r\n\r\nSolution\r\n<table id=\"eip-id1168469868726\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>1.\u00a0 Find the circumference when [latex]r={\\Large\\frac{14}{15}}[\/latex]<\/td>\r\n<td><\/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 id=\"eip-id1168467437743\" class=\"unnumbered unstyled\" summary=\".\">\r\n<tbody>\r\n<tr>\r\n<td>2.\u00a0 Find the area when [latex]r={\\Large\\frac{14}{15}}[\/latex].<\/td>\r\n<td><\/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[\/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]146611[\/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 of a circle<\/li>\n<li>Find the area of a circle<\/li>\n<\/ul>\n<\/div>\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 radius, 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 diameter. The diameter is twice as long as the radius. See the image below.<\/p>\n<p>The size of a circle can be measured in two ways. The distance around a circle is called its circumference.<\/p>\n<p><img decoding=\"async\" class=\"aligncenter\" 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.\" \/><br \/>\nArchimedes 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;). However, the exact value of [latex]\\pi[\/latex] cannot be calculated since the decimal never ends or repeats (we will learn more about numbers like this in The Properties of Real Numbers.)<\/p>\n<p>Doing the Manipulative Mathematics activity Pi Lab will help you develop a better understanding of pi.<\/p>\n<p>If we want the exact circumference or area of a circle, we leave the symbol [latex]\\pi[\/latex] in the answer. We can get an approximate answer by substituting [latex]3.14[\/latex] as the value of [latex]\\pi[\/latex]. We use the symbol [latex]\\approx[\/latex] to show that the result is approximate, not exact.<\/p>\n<div class=\"textbox shaded\">\n<h3>Properties of Circles<\/h3>\n<p><img decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/277\/2017\/04\/24221637\/CNX_BMath_Figure_05_03_010_img.png\" alt=\"A circle is shown. A line runs through the widest portion of the circle. There is a red dot at the center of the circle. The half of the line from the center of the circle to a point on the right of the circle is labeled with an r. The half of the line from the center of the circle to a point on the left of the circle is also labeled with an r. The two sections labeled r have a brace drawn underneath showing that the entire segment is labeled d.\" \/><\/p>\n<p>[latex]\\begin{array}{c}r\\text{ is the length of the radius.}\\hfill \\\\ d\\text{ is the length of the diameter.}\\hfill \\end{array}[\/latex]<br \/>\n[latex]\\begin{array}{cccc}\\text{The circumference is }2\\pi \\mathit{\\text{r}}.\\hfill & & & C=2\\pi \\mathit{\\text{r}}\\hfill \\\\ \\text{The area is }\\pi{\\mathit{\\text{r}}}^{2}.\\hfill & & & A=\\pi{\\mathit{\\text{r}}}^{2}\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\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].<\/p>\n<p>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 style=\"text-align: left\">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.<br \/>\nTo 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<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>A circle has radius [latex]10[\/latex] centimeters.Approximate its circumference and area.<\/p>\n<p>Solution<\/p>\n<table id=\"eip-id1168467143703\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.\u00a0 Find the circumference when [latex]r=10[\/latex].<\/td>\n<td><\/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]3.14[\/latex] for [latex]\\pi[\/latex] and 10 for , [latex]r[\/latex] .<\/td>\n<td>[latex]C\\approx 2\\left(3.14\\right)\\left(10\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]C\\approx 62.8\\text{ centimeters}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168469856534\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.\u00a0 Find the area when [latex]r=10[\/latex].<\/td>\n<td><\/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]3.14[\/latex] for [latex]\\pi[\/latex] and 10 for [latex]r[\/latex] .<\/td>\n<td>[latex]A\\approx \\left(3.14\\right){\\text{(}10\\text{)}}^{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]A\\approx 314\\text{ square centimeters}[\/latex]<\/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<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>A circle has radius [latex]42.5[\/latex] centimeters. Approximate its circumference and area.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q272573\">Show Solution<\/span><\/p>\n<div id=\"q272573\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168468436463\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.\u00a0 Find the circumference when [latex]r=42.5[\/latex]<\/td>\n<td><\/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]3.14[\/latex] for [latex]\\pi[\/latex] and [latex]42.5[\/latex] for [latex]r[\/latex]<\/td>\n<td>[latex]C\\approx 2\\left(3.14\\right)\\left(42.5\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Multiply.<\/td>\n<td>[latex]C\\approx 266.9\\text{ centimeters}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table id=\"eip-id1168466204784\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">2.\u00a0 Find the area when [latex]r=42.5[\/latex]<\/td>\n<td style=\"height: 15px\"><\/td>\n<\/tr>\n<tr style=\"height: 15px\">\n<td style=\"height: 15px\">Write the formula for area.<\/td>\n<td style=\"height: 15px\">[latex]A=\\pi {\\mathit{\\text{r}}}^{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 31px\">\n<td style=\"height: 31px\">Substitute [latex]3.14[\/latex] for [latex]\\pi[\/latex] and [latex]42.5[\/latex] for [latex]r[\/latex] .<\/td>\n<td style=\"height: 31px\">[latex]A\\approx \\left(3.14\\right){\\text{(}42.5\\text{)}}^{2}[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 31px\">\n<td style=\"height: 31px\">Multiply.<\/td>\n<td style=\"height: 31px\">[latex]A\\approx 5671.625\\text{ square centimeters}[\/latex]<\/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=\"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>Watch the following video to see another example of how to find the circumference of a circle.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" 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<p>In the next video example, we find the area of a circle.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" 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<h2>Approximate [latex]\\pi[\/latex] with a Fraction<\/h2>\n<p>Convert the fraction [latex]{\\Large\\frac{22}{7}}[\/latex] to a decimal. 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]. And, since [latex]{\\Large\\frac{22}{7}}[\/latex] is also an approximation of [latex]\\pi[\/latex], we will use the [latex]\\approx[\/latex] symbol to show we have an approximate value.<\/p>\n<div class=\"textbox exercises\">\n<h3>example<\/h3>\n<p>A circle has radius [latex]{\\Large\\frac{14}{15}}[\/latex] meters. Approximate its circumference and area.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q418144\">Show Solution<\/span><\/p>\n<div id=\"q418144\" class=\"hidden-answer\" style=\"display: none\">\n<p>Solution<\/p>\n<table id=\"eip-id1168469868726\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>1.\u00a0 Find the circumference when [latex]r={\\Large\\frac{14}{15}}[\/latex]<\/td>\n<td><\/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 id=\"eip-id1168467437743\" class=\"unnumbered unstyled\" summary=\".\">\n<tbody>\n<tr>\n<td>2.\u00a0 Find the area when [latex]r={\\Large\\frac{14}{15}}[\/latex].<\/td>\n<td><\/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<\/div>\n<p>&nbsp;<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>try it<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm146611\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=146611&theme=oea&iframe_resize_id=ohm146611&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-4436\">\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 146611, 146564, 146563. <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>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/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":25777,"menu_order":1,"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\":\"cc\",\"description\":\"Examples: Determine the Circumference of a Circle\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/sHtsnC2Mgnk\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Example: Determine the Area of a Circle\",\"author\":\"James Sousa (Mathispower4u.com)\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/SIKkWLqt2mQ\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Question ID 146611, 146564, 146563\",\"author\":\"Lumen Learning\",\"organization\":\"\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"IMathAS Community License CC-BY + GPL\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-4436","chapter","type-chapter","status-web-only","hentry"],"part":3706,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4436","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":1,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4436\/revisions"}],"predecessor-version":[{"id":4437,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4436\/revisions\/4437"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/3706"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/4436\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=4436"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=4436"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=4436"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=4436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}