{"id":1794,"date":"2025-07-28T19:39:48","date_gmt":"2025-07-28T19:39:48","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&#038;p=1794"},"modified":"2025-08-13T03:02:43","modified_gmt":"2025-08-13T03:02:43","slug":"arcs-and-sectors-learn-it-1","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/arcs-and-sectors-learn-it-1\/","title":{"raw":"Arcs and Sectors: Learn It 1","rendered":"Arcs and Sectors: Learn It 1"},"content":{"raw":"<section class=\"citations-section focusable\" role=\"contentinfo\"><section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\r\n<ul>\r\n \t<li>Find the length of a circular arc.<\/li>\r\n \t<li>Find the area of a sector of a circle.<\/li>\r\n \t<li>Use linear and angular speed to describe motion on a circular path.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Determining the Length of an Arc<\/h2>\r\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>radian measure<\/strong> [latex]\\theta [\/latex] of an angle was defined as the ratio of the <strong>arc length<\/strong> [latex]s[\/latex] of a circular arc to the radius [latex]r[\/latex] of the circle, [latex]\\theta =\\frac{s}{r}[\/latex]<\/section><\/section>From this relationship, we can find arc length along a circle, given an angle.\r\n\r\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\r\n<h3>arc length on a circle<\/h3>\r\nIn a circle of radius <em>r<\/em>, the length of an arc [latex]s[\/latex] subtended by an angle with measure [latex]\\theta [\/latex] in radians is <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; text-align: center;\">[latex]s=r\\theta [\/latex]<\/span>\r\n<div style=\"text-align: center;\">\r\n\r\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003450\/CNX_Precalc_Figure_05_01_024F2.jpg\" alt=\"Illustration of circle with angle theta, radius r, and arc with length s.\" width=\"349\" height=\"348\" \/>\r\n\r\n<\/div>\r\n<\/section><section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a circle of radius [latex]r[\/latex], calculate the length [latex]s[\/latex] of the arc subtended by a given angle of measure [latex]\\theta [\/latex].<\/strong>\r\n<ol>\r\n \t<li>If necessary, convert [latex]\\theta [\/latex] to radians.<\/li>\r\n \t<li>Multiply the radius [latex]r[\/latex] by the radian measure of [latex]\\theta :s=r\\theta [\/latex].<\/li>\r\n<\/ol>\r\n<\/section><section class=\"textbox example\" aria-label=\"Example\">Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.\r\n<ol>\r\n \t<li>In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?<\/li>\r\n \t<li>Use your answer from part (a) to determine the radian measure for Mercury\u2019s movement in one Earth day.<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"107850\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"107850\"]\r\n<ol>\r\n \t<li>Let\u2019s begin by finding the circumference of Mercury\u2019s orbit.\r\n<div style=\"text-align: center;\">[latex]\\begin{align}C&amp;=2\\pi r \\\\ &amp;=2\\pi \\left(36\\text{ million miles}\\right) \\\\ &amp;\\approx 226\\text{ million miles} \\end{align}[\/latex]<\/div>\r\nSince Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:\r\n<div style=\"text-align: center;\">[latex]\\left(0.0114\\right)226\\text{ million miles = 2}\\text{.58 million miles}[\/latex]<\/div><\/li>\r\n \t<li>Now, we convert to radians:\r\n<div style=\"text-align: center;\">[latex]\\begin{align}\\text{radian}&amp;=\\frac{\\text{arclength}}{\\text{radius}} \\\\ &amp;=\\frac{2.\\text{58 million miles}}{36\\text{ million miles}} \\\\ &amp;=0.0717 \\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">Find the arc length along a circle of radius 10 units subtended by an angle of 215\u00b0.[reveal-answer q=\"188778\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"188778\"][latex]\\frac{215\\pi }{18}=37.525\\text{ units}[\/latex][\/hidden-answer]<\/section><section class=\"textbox tryIt\" aria-label=\"Try It\">[ohm_question hide_question_numbers=1]172921[\/ohm_question]<\/section>","rendered":"<section class=\"citations-section focusable\" role=\"contentinfo\">\n<section class=\"textbox learningGoals\" aria-label=\"Learning Goals\">\n<ul>\n<li>Find the length of a circular arc.<\/li>\n<li>Find the area of a sector of a circle.<\/li>\n<li>Use linear and angular speed to describe motion on a circular path.<\/li>\n<\/ul>\n<\/section>\n<h2>Determining the Length of an Arc<\/h2>\n<section class=\"textbox recall\" aria-label=\"Recall\">The <strong>radian measure<\/strong> [latex]\\theta[\/latex] of an angle was defined as the ratio of the <strong>arc length<\/strong> [latex]s[\/latex] of a circular arc to the radius [latex]r[\/latex] of the circle, [latex]\\theta =\\frac{s}{r}[\/latex]<\/section>\n<\/section>\n<p>From this relationship, we can find arc length along a circle, given an angle.<\/p>\n<section class=\"textbox keyTakeaway\" aria-label=\"Key Takeaway\">\n<h3>arc length on a circle<\/h3>\n<p>In a circle of radius <em>r<\/em>, the length of an arc [latex]s[\/latex] subtended by an angle with measure [latex]\\theta[\/latex] in radians is <span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif; text-align: center;\">[latex]s=r\\theta[\/latex]<\/span><\/p>\n<div style=\"text-align: center;\">\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27003450\/CNX_Precalc_Figure_05_01_024F2.jpg\" alt=\"Illustration of circle with angle theta, radius r, and arc with length s.\" width=\"349\" height=\"348\" \/><\/p>\n<\/div>\n<\/section>\n<section class=\"textbox questionHelp\" aria-label=\"Question Help\"><strong>How To: Given a circle of radius [latex]r[\/latex], calculate the length [latex]s[\/latex] of the arc subtended by a given angle of measure [latex]\\theta[\/latex].<\/strong><\/p>\n<ol>\n<li>If necessary, convert [latex]\\theta[\/latex] to radians.<\/li>\n<li>Multiply the radius [latex]r[\/latex] by the radian measure of [latex]\\theta :s=r\\theta[\/latex].<\/li>\n<\/ol>\n<\/section>\n<section class=\"textbox example\" aria-label=\"Example\">Assume the orbit of Mercury around the sun is a perfect circle. Mercury is approximately 36 million miles from the sun.<\/p>\n<ol>\n<li>In one Earth day, Mercury completes 0.0114 of its total revolution. How many miles does it travel in one day?<\/li>\n<li>Use your answer from part (a) to determine the radian measure for Mercury\u2019s movement in one Earth day.<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q107850\">Show Solution<\/button><\/p>\n<div id=\"q107850\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Let\u2019s begin by finding the circumference of Mercury\u2019s orbit.\n<div style=\"text-align: center;\">[latex]\\begin{align}C&=2\\pi r \\\\ &=2\\pi \\left(36\\text{ million miles}\\right) \\\\ &\\approx 226\\text{ million miles} \\end{align}[\/latex]<\/div>\n<p>Since Mercury completes 0.0114 of its total revolution in one Earth day, we can now find the distance traveled:<\/p>\n<div style=\"text-align: center;\">[latex]\\left(0.0114\\right)226\\text{ million miles = 2}\\text{.58 million miles}[\/latex]<\/div>\n<\/li>\n<li>Now, we convert to radians:\n<div style=\"text-align: center;\">[latex]\\begin{align}\\text{radian}&=\\frac{\\text{arclength}}{\\text{radius}} \\\\ &=\\frac{2.\\text{58 million miles}}{36\\text{ million miles}} \\\\ &=0.0717 \\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\">Find the arc length along a circle of radius 10 units subtended by an angle of 215\u00b0.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q188778\">Show Solution<\/button><\/p>\n<div id=\"q188778\" class=\"hidden-answer\" style=\"display: none\">[latex]\\frac{215\\pi }{18}=37.525\\text{ units}[\/latex]<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\" aria-label=\"Try It\"><iframe loading=\"lazy\" id=\"ohm172921\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=172921&theme=lumen&iframe_resize_id=ohm172921&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":13,"menu_order":14,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":178,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1794"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":5,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1794\/revisions"}],"predecessor-version":[{"id":2417,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1794\/revisions\/2417"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/178"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/1794\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=1794"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=1794"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=1794"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=1794"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}