{"id":87631,"date":"2021-08-09T14:26:15","date_gmt":"2021-08-09T19:26:15","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=87631"},"modified":"2026-03-26T10:08:16","modified_gmt":"2026-03-26T15:08:16","slug":"arcs-and-angles-of-circles","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/arcs-and-angles-of-circles\/","title":{"rendered":"Arcs and Angles of Circles"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_Yhl0zX1uQ-0\" style=\"position: relative;\">\n\t\t\t<picture>\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.webp\" type=\"image\/webp\">\n\t\t\t\t<source srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" type=\"image\/jpeg\"> \n\t\t\t\t<img fetchpriority=\"high\" decoding=\"async\" loading=\"eager\" id=\"videoThumbnailImage_Yhl0zX1uQ-0\" data-source-videoID=\"Yhl0zX1uQ-0\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Arcs and Angles of Circles Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Arcs and Angles of Circles\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_Yhl0zX1uQ-0:hover {cursor:pointer;} img#videoThumbnailImage_Yhl0zX1uQ-0 {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1736-thumb-final-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_Yhl0zX1uQ-0\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_Yhl0zX1uQ-0\" style=\"display: none;position: absolute;top: -24px;width: 100%;text-align: center;\"><span style=\"font-style: italic;font-size: small;border-top: 1px solid #fc0;\">Having trouble? <a href=\"https:\/\/www.youtube.com\/watch?v='+videoId+'\" target=\"_blank\">Click here to watch on YouTube.<\/a><\/span><\/div>';\n\t\t\t\tlet tag = document.createElement(\"iframe\");\n\t\t\t\ttag.id = \"yt\" + videoId;\n\t\t\t\ttag.src = \"https:\/\/www.youtube-nocookie.com\/embed\/\" + videoId + \"?autoplay=1&controls=1&wmode=opaque&rel=0&egm=0&iv_load_policy=3&hd=0&enablejsapi=1\";\n\t\t\t\ttag.frameborder = 0;\n\t\t\t\ttag.allow = \"autoplay; fullscreen\";\n\t\t\t\ttag.width = this.width;\n\t\t\t\ttag.height = this.height;\n\t\t\t\ttag.setAttribute(\"data-matomo-title\",\"Arcs and Angles of Circles\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_Yhl0zX1uQ-0\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_Yhl0zX1uQ-0\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_Yhl0zX1uQ-0\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction j0M_Function() {\n  var x = document.getElementById(\"j0M\");\n  if (x.style.display === \"none\") {\n    x.style.display = \"block\";\n  } else {\n    x.style.display = \"none\";\n  }\n}\n<\/script><\/p>\n<div class=\"moc-toc hide-on-desktop hide-on-tablet\">\n<div><button onclick=\"j0M_Function()\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2024\/12\/toc2.svg\" width=\"16\" height=\"16\" alt=\"show or hide table of contents\"><\/button><\/p>\n<p>On this page<\/p>\n<\/div>\n<nav id=\"j0M\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_is_an_Arc\" class=\"smooth-scroll\">What is an Arc?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Chords,_Secants,_and_Tangents\" class=\"smooth-scroll\">Chords, Secants, and Tangents<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Inscribed_Angle_Theorem\" class=\"smooth-scroll\">Inscribed Angle Theorem<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Central_Angle_Theorem\" class=\"smooth-scroll\">Central Angle Theorem<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Arcs_and_Angles_of_Circles_Practice_Questions\" class=\"smooth-scroll\">Arcs and Angles of Circles Practice Questions<\/a><\/li>\n<\/ul>\n<\/nav>\n<\/div>\n<div class=\"accordion\"><input id=\"transcript\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"transcript\">Transcript<\/label><input id=\"PQs\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQs\">Practice<\/label>\n<div class=\"spoiler\" id=\"transcript-spoiler\">\n<p>Hello, and welcome to this video about arcs and angles of circles! In this video, we will explore the different parts of circles and how to use them to solve problems.<\/p>\n<p>Let\u2019s learn about arcs and angles of circles!<\/p>\n<h2><span id=\"What_is_an_Arc\" class=\"m-toc-anchor\"><\/span>What is an Arc?<\/h2>\n<p>\nWhen we look around us, after polygons, circles are the next most common shape that we are surrounded by in our environment. A circle is created by a 360\u00b0 rotation. <\/p>\n<p>An <strong>arc<\/strong> of the circle is a part of the circle, and we identify arcs using points on the circle.<\/p>\n<p>Arc <span style=\"font-style:normal; font-size:90%\">\\(RS\\)<\/span> is the curved part of circle <span style=\"font-style:normal; font-size:90%\">\\(A\\)<\/span> shown with the purple mark. Because arc <span style=\"font-style:normal; font-size:90%\">\\(RS\\)<\/span> is less than 180\u00b0, we call this arc a <strong>minor arc<\/strong>. Arc <span style=\"font-style:normal; font-size:90%\">\\(RTS\\)<\/span>, or <span style=\"font-style:normal; font-size:90%\">\\(STR\\)<\/span>, is a <strong>major arc<\/strong> because it is greater than 180\u00b0.<\/p>\n<h3><span id=\"Finding_the_Measure_of_an_Arc\" class=\"m-toc-anchor\"><\/span>Finding the Measure of an Arc<\/h3>\n<p>\nWe can find the measure of an arc using the fact that a circle is 360\u00b0. If we know that the measure of arc <span style=\"font-style:normal; font-size:90%\">\\(RS\\)<\/span> is 130\u00b0, we can find the measure of arc <span style=\"font-style:normal; font-size:90%\">\\(RTS\\)<\/span> by subtracting the measure of arc <span style=\"font-style:normal; font-size:90%\">\\(RS\\)<\/span> from 360\u00b0, which would make the measure of arc <span style=\"font-style:normal; font-size:90%\">\\(RTS\\)<\/span> 230\u00b0.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89983\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-1.png\" alt=\"Image showing that a minor arc is less than 180 degrees, and a major arc is greater than 180. \" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-1.png 1917w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-1-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-1-1024x577.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-1-768x433.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-1-1536x865.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>Some important lines and segments associated with circles are chords, secants, and tangents.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-2.png\" alt=\"Image showing Tangents, Secants, and Chords. \" width=\"777\" height=\"437\" class=\"aligncenter size-full wp-image-89989\" style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-2.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-2-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-2-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-2-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-2-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<h2><span id=\"Chords,_Secants,_and_Tangents\" class=\"m-toc-anchor\"><\/span>Chords, Secants, and Tangents<\/h2>\n<p>\nA <strong>chord<\/strong> is a line segment with either end touching the circle. Line segment <span style=\"font-style:normal; font-size:90%\">\\(LM\\)<\/span> is a chord on circle <span style=\"font-style:normal; font-size:90%\">\\(A\\)<\/span>.<\/p>\n<p>A <strong>secant line<\/strong> is similar to a chord, except it is a line that passes through two points on the circle instead of being a line segment.<\/p>\n<p>And a <strong>tangent line<\/strong> is a very special kind of line that only touches the circle at one point, called the <strong>point of tangency<\/strong>. Line <span style=\"font-style:normal; font-size:90%\">\\(PQ\\)<\/span> is a tangent and point <span style=\"font-style:normal; font-size:90%\">\\(Q\\)<\/span> is the point of tangency.<\/p>\n<p>Knowing these vocabulary words related to circles is important in helping understand the circle theorems which will help us solve problems.<\/p>\n<p>The radius of the circle is always perpendicular to the point of tangency. <\/p>\n<h2><span id=\"Inscribed_Angle_Theorem\" class=\"m-toc-anchor\"><\/span>Inscribed Angle Theorem<\/h2>\n<p>\nWhen two chords share a point on the circle, an <strong>inscribed angle<\/strong> is formed. In circle <span style=\"font-style:normal; font-size:90%\">\\(A\\)<\/span>, the chords <span style=\"font-style:normal; font-size:90%\">\\(RS\\)<\/span> and <span style=\"font-style:normal; font-size:90%\">\\(TS\\)<\/span> form the inscribed angle <span style=\"font-style:normal; font-size:90%\">\\(RST\\)<\/span>, which has its vertex on the circle.<\/p>\n<p>The arc that is formed by the legs of angle <span style=\"font-style:normal; font-size:90%\">\\(RST\\)<\/span> is called an <strong>intercepted arc<\/strong>.<\/p>\n<p>An angle that has the center of the circle as its vertex is naturally called a <strong>central angle<\/strong>. In circle <span style=\"font-style:normal; font-size:90%\">\\(C\\)<\/span>, angle <span style=\"font-style:normal; font-size:90%\">\\(MCN\\)<\/span> is a central angle and arc <span style=\"font-style:normal; font-size:90%\">\\(MN\\)<\/span> is its intercepted arc. <\/p>\n<p>If we are given the measure of the inscribed angle or the intercepted arc, we will be able to find the measure of the other using a circle theorem that tells us that the measure of an inscribed angle is one-half the measure of its intercepted arc. Or we could say that the intercepted arc is twice as long as the inscribed angle.<\/p>\n<p>For example, in circle <span style=\"font-style:normal; font-size:90%\">\\(A\\)<\/span>, the measure of inscribed angle <span style=\"font-style:normal; font-size:90%\">\\(RST\\)<\/span> is given to us as 60\u00b0. According to the <strong>inscribed angle theorem<\/strong>, the measure of arc <span style=\"font-style:normal; font-size:90%\">\\(RT\\)<\/span> is <span style=\"font-style:normal; font-size:90%\">\\(60\u00b0 \\times 2\\)<\/span>, which is 120\u00b0. <\/p>\n<p>Here is a special case of an inscribed angle.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-Circles-3.png\" alt=\"Image of a unique inscribed PQRA angle with an arc of 90 degrees.\" width=\"777\" height=\"437\" class=\"aligncenter size-full wp-image-89992\"style=\"box-shadow: 1.5px 1.5px 3px grey\" \/><\/p>\n<p>In circle <span style=\"font-style:normal; font-size:90%\">\\(A\\)<\/span>, inscribed angle <span style=\"font-style:normal; font-size:90%\">\\(PQR\\)<\/span> encompasses the diameter of the circle, line segment <span style=\"font-style:normal; font-size:90%\">\\(PR\\)<\/span> Remember, the <strong>diameter<\/strong> of the circle divides the circle into two equal parts, called <strong>semicircles<\/strong>.<\/p>\n<p>A circle has a total of 360\u00b0, so a semicircle has 180\u00b0, or half the measure of a full circle. So this must mean that the intercepted arc of angle <span style=\"font-style:normal; font-size:90%\">\\(PQR\\)<\/span> is 180\u00b0. If we use the inscribed angle theorem, we find out that angle <span style=\"font-style:normal; font-size:90%\">\\(PQR\\)<\/span> is 90\u00b0 because it is half the degree measure of the intercepted arc.<\/p>\n<h2><span id=\"Central_Angle_Theorem\" class=\"m-toc-anchor\"><\/span>Central Angle Theorem<\/h2>\n<p>\nAnother important theorem we are going to take a look at is the central angle theorem. This theorem says that the measure of a central angle is equal to the measure of its intercepted arc. In circle <span style=\"font-style:normal; font-size:90%\">\\(C\\)<\/span>, since the measure of angle <span style=\"font-style:normal; font-size:90%\">\\(MCN\\)<\/span>, the central angle, is 110\u00b0, then arc <span style=\"font-style:normal; font-size:90%\">\\(MN\\)<\/span>, its intercepted arc, is also 110\u00b0.<\/p>\n<p>Let\u2019s take a look at an example problem. Line <span style=\"font-style:normal; font-size:90%\">\\(LM\\)<\/span> is tangent to circle <span style=\"font-style:normal; font-size:90%\">\\(T\\)<\/span> at point <span style=\"font-style:normal; font-size:90%\">\\(L\\)<\/span>. The measure of the central angle <span style=\"font-style:normal; font-size:90%\">\\(T\\)<\/span> is 35\u00b0. What is the measure of angle <span style=\"font-style:normal; font-size:90%\">\\(LMT\\)<\/span>? <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131852403.png\" alt=\"\" width=\"597\" height=\"524\" class=\"aligncenter size-full wp-image-205613\" style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131852403.png 1194w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131852403-300x263.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131852403-1024x899.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131852403-768x674.png 768w\" sizes=\"auto, (max-width: 597px) 100vw, 597px\" \/><\/p>\n<p>Since we know that the sum of the interior angles of any triangle is 180\u00b0 and that the radius is always perpendicular to the point of tangency, we know that angle <span style=\"font-style:normal; font-size:90%\">\\(TLM\\)<\/span> is 90\u00b0. We can subtract <span style=\"font-style:normal; font-size:90%\">\\((90\u00b0+35\u00b0)\\)<\/span> from 180\u00b0 to find the measure of angle <span style=\"font-style:normal; font-size:90%\">\\(LMT\\)<\/span>. <span style=\"font-style:normal; font-size:90%\">\\(180 \u2013 (90+35) = 55\\)<\/span>, therefore the measure of angle <span style=\"font-style:normal; font-size:90%\">\\(LMT\\)<\/span> is 55\u00b0.<\/p>\n<p>When secants intersect inside a circle, we can find the measure of the vertical angles created by using the measure of the arcs formed by the angles. It also works the other way around, if we need to find the measure of the arcs using the measure of the angles. <\/p>\n<p>Let\u2019s take a look at circle <span style=\"font-style:normal; font-size:90%\">\\(K\\)<\/span>. <\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131224565.png\" alt=\"\" width=\"595.5\" height=\"432.5\" class=\"aligncenter size-full wp-image-205601\" style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131224565.png 1191w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131224565-300x218.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131224565-1024x744.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131224565-768x558.png 768w\" sizes=\"(max-width: 1191px) 100vw, 1191px\" \/><\/p>\n<p>The secant lines <span style=\"font-style:normal; font-size:90%\">\\(DE\\)<\/span> and <span style=\"font-style:normal; font-size:90%\">\\(FG\\)<\/span> intersect at point <span style=\"font-style:normal; font-size:90%\">\\(H\\)<\/span> and create the arcs <span style=\"font-style:normal; font-size:90%\">\\(DF\\)<\/span>, <span style=\"font-style:normal; font-size:90%\">\\(FE\\)<\/span>, <span style=\"font-style:normal; font-size:90%\">\\(EG\\)<\/span>, and <span style=\"font-style:normal; font-size:90%\">\\(GD\\)<\/span>. Find the measure of any of the vertical angles (notice that there are two sets) by adding together the measures of the intercepted arcs and dividing by 2.<\/p>\n<p>We will use the formulas <span style=\"font-style:normal; font-size:90%\">\\(m\\angle DHF=\\frac{1}{2}\\)<\/span> (arc <span style=\"font-style:normal; font-size:90%\">\\(DF\\)<\/span> + arc <span style=\"font-style:normal; font-size:90%\">\\(GE\\)<\/span>) and <span style=\"font-style:normal; font-size:90%\">\\(m\\angle FHE=\\frac{1}{2}\\)<\/span> (arc <span style=\"font-style:normal; font-size:90%\">\\(FE\\)<\/span> + arc <span style=\"font-style:normal; font-size:90%\">\\(DG\\)<\/span>). Then we will use the vertical rule theorem to find <span style=\"font-style:normal; font-size:90%\">\\(m\\angle EHG\\)<\/span> and <span style=\"font-style:normal; font-size:90%\">\\(m\\angle GHD\\)<\/span>. <\/p>\n<p>Let\u2019s practice. The measure of arc <span style=\"font-style:normal; font-size:90%\">\\(GE\\)<\/span> is 170\u00b0, arc <span style=\"font-style:normal; font-size:90%\">\\(GD\\)<\/span> is 80\u00b0, arc <span style=\"font-style:normal; font-size:90%\">\\(DF\\)<\/span> is 60\u00b0, and arc <span style=\"font-style:normal; font-size:90%\">\\(FE\\)<\/span> is 50\u00b0. To find the <span style=\"font-style:normal; font-size:90%\">\\(m\\angle DHF\\)<\/span>, we will add the intercepted arcs, <span style=\"font-style:normal; font-size:90%\">\\(DF\\)<\/span> and <span style=\"font-style:normal; font-size:90%\">\\(GE\\)<\/span>, then multiply by <span style=\"font-style:normal; font-size:90%\">\\(\\frac{1}{2}\\)<\/span>. Therefore, <span style=\"font-style:normal; font-size:90%\">\\(m\\angle DHF=\\frac{1}{2}(60\u00b0 +170 \u00b0)\\)<\/span>. So <span style=\"font-style:normal; font-size:90%\">\\(\\angle DHF=115\u00b0 \\)<\/span>. We will use the equation, <span style=\"font-style:normal; font-size:90%\">\\(m\\angle FHE=\\frac{1}{2}(50\u00b0 +80\u00b0)\\)<\/span> to find the measure of <span style=\"font-style:normal; font-size:90%\">\\(\\angle FHE\\)<\/span>, which is <span style=\"font-style:normal; font-size:90%\">\\(m\\angle FHE=65\u00b0\\)<\/span>.<\/p>\n<p>When two secant lines intersect outside the circle, we use a different method to find the measure of the angles. When this happens, you look at the two intercepted arcs created by the angle and subtract the measure of the smaller arc from the measure of the larger arc, then multiply by <span style=\"font-style:normal; font-size:90%\">\\(\\frac{1}{2}\\)<\/span>.<\/p>\n<p><img decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131634799.png\" alt=\"\" width=\"547.6\" height=\"504.4\" class=\"aligncenter size-full wp-image-205607\" style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131634799.png 1369w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131634799-300x276.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131634799-1024x943.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/10\/image_2023-10-23_131634799-768x707.png 768w\" sizes=\"(max-width: 1369px) 100vw, 1369px\" \/><\/p>\n<p>We want to find the <span style=\"font-style:normal; font-size:90%\">\\(m\\angle WYU\\)<\/span> that is created by the intersection of secants <span style=\"font-style:normal; font-size:90%\">\\(WX\\)<\/span> and <span style=\"font-style:normal; font-size:90%\">\\(UV\\)<\/span> on circle <span style=\"font-style:normal; font-size:90%\">\\(Z\\)<\/span>. <span style=\"font-style:normal; font-size:90%\">\\(\\angle WYU\\)<\/span> is intercepted by arcs <span style=\"font-style:normal; font-size:90%\">\\(XV\\)<\/span> and <span style=\"font-style:normal; font-size:90%\">\\(WU\\)<\/span>. The measure of arc <span style=\"font-style:normal; font-size:90%\">\\(XV\\)<\/span> is 30\u00b0; the measure of arc <span style=\"font-style:normal; font-size:90%\">\\(WU\\)<\/span> is 80\u00b0. To find the measure of angle <span style=\"font-style:normal; font-size:90%\">\\(WYU\\)<\/span>, we use the formula: <span style=\"font-style:normal; font-size:90%\">\\(m\\angle WYU= \\frac{1}{2}\\)<\/span> (arc <span style=\"font-style:normal; font-size:90%\">\\(WU\\)<\/span> \u2013 arc <span style=\"font-style:normal; font-size:90%\">\\(XV\\)<\/span>), therefore, <span style=\"font-style:normal; font-size:90%\">\\(m\\angle WYU=\\frac{1}{2}(80\u00b0-30\u00b0) \\text{, }m\\angle WYU=25\u00b0\\)<\/span>.<\/p>\n<p>A sector of a circle is a section of a circle between two radii. The red area in the diagram is an example of a sector. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-circles-4.png\" alt=\"Image of a sector: a section of a circle between two radii. This sector has an arc of 70 degrees. \" width=\"777\" height=\"437\" class=\"aligncenter size-full wp-image-89995\" style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-circles-4.png 1917w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-circles-4-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-circles-4-1024x577.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-circles-4-768x433.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Arcs-and-Angles-on-circles-4-1536x865.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>We can find the length of an arc if we are given the length of the radius and the measure of the central angle using the formula <span style=\"font-style:normal; font-size:90%\">\\(s=\\frac{\\pi r\\theta }{180\u00b0}\\)<\/span>, where <span style=\"font-style:normal; font-size:90%\">\\(s\\)<\/span> is the arc length, <span style=\"font-style:normal; font-size:90%\">\\(r\\)<\/span> is the length of the radius and <span style=\"font-style:normal; font-size:90%\">\\(\\theta \\)<\/span> is the measure of the central angle. In circle A, the length of the radius is 5 cm, and the measure of angle <span style=\"font-style:normal; font-size:90%\">\\(MAN\\)<\/span> is 70\u00b0. We want to find the length of arc <span style=\"font-style:normal; font-size:90%\">\\(MN\\)<\/span>. We will start by substituting what we have into the formula, <span style=\"font-style:normal; font-size:90%\">\\(s=\\frac{\\pi (5)(70)}{180\u00b0}\\)<\/span>. We will simplify to get <span style=\"font-style:normal; font-size:90%\">\\(s=6.11\\text{ cm}\\)<\/span> , which is the length of arc <span style=\"font-style:normal; font-size:90%\">\\(MN\\)<\/span>.<\/p>\n<p>Thanks for watching this video, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Arcs_and_Angles_of_Circles_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Arcs and Angles of Circles Practice Questions<\/h2>\n\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #1:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nIf \\(\\angle B\\) is 21\u00b0, what is the measure of arc \\(AC\\)? <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Arcs-and-Angles-Example-1.svg\" alt=\"A circle with points A, B, and C on its circumference. Lines connect B to A and B to C. Arc AC is highlighted in red.\" width=\"276\" height=\"276\" class=\"aligncenter size-full wp-image-287513\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">82\u00b0<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-2\">42\u00b0<\/div><div class=\"PQ\"  id=\"PQ-1-3\">165\u00b0<\/div><div class=\"PQ\"  id=\"PQ-1-4\">21\u00b0<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-1\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-1-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>According to the inscribed angle theorem, arc \\(AC \\) will be twice as large as the inscribed \\(\\angle B\\). This means that arc \\(AC\\) will be \\(2 \\times 21\u00b0=42\u00b0\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-1-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-1-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #2:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nLine \\(\\overline{LM}\\) is tangent to circle \\(J\\) at point \\(L\\). If \\(\\angle J\\) is 27\u00b0, what is the measure of \\(\\angle M\\)? <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Arcs-and-Angles-Example-2.svg\" alt=\"A circle with center J, tangent line at point L, points K and M on the tangent, and an angle marked at KLM.\" width=\"344\" height=\"279\" class=\"aligncenter size-full wp-image-287501\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">65\u00b0<\/div><div class=\"PQ\"  id=\"PQ-2-2\">83\u00b0<\/div><div class=\"PQ\"  id=\"PQ-2-3\">44\u00b0<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-4\">63\u00b0<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-2\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-2-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Since line \\(\\overline{LM}\\) is tangent to circle \\(J\\) at point \\(L\\), the line \\(\\overline{JL}\\) and \\(\\overline{LM}\\) form a right angle. This means that \\(\\angle L\\) is 90\u00b0.<\/p>\n<p>\\(\\angle J\\) is 27\u00b0, so add \\(\\angle J\\) and \\(\\angle L\\) to get \\(90+27=117\\).<\/p>\n<p>Since there are 180\u00b0 in a triangle, subtract 117 from 180 to get \\(180-117=63\\).<\/p>\n<p>Therefore, the measure of \\(\\angle M\\) is 63\u00b0. <\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-2-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-2-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #3:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nArc \\(BD\\) is 15\u00b0. Arc \\(AE\\) is 48\u00b0. Find the measure of angle \\(ACE\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Arcs-and-Angles-Example-3.svg\" alt=\"Two intersecting straight lines pass through points A, B, D, and E, with B and D on the circumference of a circle. The lines meet at point C outside the circle.\" width=\"594\" height=\"289\" class=\"aligncenter size-full wp-image-287504\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-3-1\">19.5\u00b0<\/div><div class=\"PQ correct_answer\"  id=\"PQ-3-2\">16.5\u00b0<\/div><div class=\"PQ\"  id=\"PQ-3-3\">22.5\u00b0<\/div><div class=\"PQ\"  id=\"PQ-3-4\">19\u00b0<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-3\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-3-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Look at the two intercepted arcs. \\(\\angle ACE\\) will be half the difference between the arc lengths of these intercepted arcs.<\/p>\n<p>Subtract the smaller arc from the larger arc. In this case, subtract arc \\(BD\\) from arc \\(AE\\) to get \\(48-15=33\\).<\/p>\n<p>Now multiply this by \\(\\frac{1}{2}\\) to get \\(33 \\times \\frac{1}{2}=16.5\u00b0\\).<\/p>\n<p>Therefore, \\(\\angle ACE\\) is 16.5\u00b0.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-3-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-3-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #4:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nYou are walking around a circular pond from one star to the other. If the radius of the pond is 8 feet, what distance have you walked? Use 3.14 for \\(\\pi\\).<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Arcs-and-Angles-Example-4.svg\" alt=\"A circle with a central angle of 120 degrees marked, and two red stars indicating the arc subtended by the angle.\" width=\"280\" height=\"271\" class=\"aligncenter size-full wp-image-287507\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">22.8 feet<\/div><div class=\"PQ\"  id=\"PQ-4-2\">19 feet<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-3\">16.78 feet<\/div><div class=\"PQ\"  id=\"PQ-4-4\">34.5 feet<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-4\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-4-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>Find the measure of the arc by using the formula \\(s=\\frac{\\pi r \\theta}{180\u00b0}\\)<\/p>\n<ul style=\"list-style-type: none\">\n<li>\\(s\\) represents the arc length.<\/li>\n<li>\\(\\pi\\) is approximated to 3.14.<\/li>\n<li>\\(r\\) is the radius.<\/li>\n<li>\\(\\theta\\) is the central angle measure.<\/li>\n<\/ul>\n<p>The formula becomes \\(s=\\frac{(3.14)(8)(120\u00b0)}{180\u00b0}=16.746\\), which simplifies to \\(s=16.7\\text{ ft}\\).<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-4-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-4-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div>\n\t\t\t\t<div class=\"PQ\">\n\t\t\t\t\t<strong>Question #5:<\/strong>\n\t\t\t\t\t<div style=\"margin-left:10px;\"><p>&nbsp;<br \/>\nA flashlight shines across a circular field. What is the angle measure of the intercepted arc that will be illuminated?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2026\/02\/Arcs-and-Angles-Example-5.svg\" alt=\"A flashlight shines a 52-degree cone of light onto a circular surface, highlighting the illuminated area inside the circle.\" width=\"346\" height=\"253\" class=\"aligncenter size-full wp-image-287510\"  role=\"img\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-5-1\">104\u00b0<\/div><div class=\"PQ\"  id=\"PQ-5-2\">120\u00b0<\/div><div class=\"PQ\"  id=\"PQ-5-3\">106\u00b0<\/div><div class=\"PQ\"  id=\"PQ-5-4\">130\u00b0<\/div>\n\t\t\t\t\t<\/div>\n\t\t\t\t\t<input id=\"PQ-5\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5\" style=\"width: 150px;\">Show Answer<\/label>\n\t\t\t\t\t<div class=\"answer\" id=\"PQ-5-spoiler\">\n\t\t\t\t\t\t<strong>Answer:<\/strong><div style=\"margin-left:10px;\"><p>The inscribed angle is 52\u00b0. The intercepted arc will be twice as much as the inscribed angle.<\/p>\n<p style=\"text-align: center\">\\(52\\times 2=104\\)<\/p>\n<p>Therefore, the angle measure that is illuminated across the field is 104\u00b0.<\/p>\n<\/div>\n\t\t\t\t\t\t<input id=\"PQ-5-hide\" type=\"checkbox\" class=\"spoiler_button\" \/><label for=\"PQ-5-hide\" style=\"width: 150px;\">Hide Answer<\/label>\n\t\t\t\t\t<\/div>\n\t\t\t\t<\/div><\/div>\n<\/div>\n\n<div class=\"home-buttons\">\n<p><a href=\"https:\/\/www.mometrix.com\/academy\/geometry\/\">Return to Geometry Videos<\/a><\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Return to Geometry Videos<\/p>\n","protected":false},"author":1,"featured_media":100741,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-87631","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-circle-video","7":"page_type-video","8":"content_type-practice-questions","9":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/87631","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/comments?post=87631"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/87631\/revisions"}],"predecessor-version":[{"id":246814,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/87631\/revisions\/246814"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100741"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=87631"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}