{"id":86488,"date":"2021-07-21T15:29:16","date_gmt":"2021-07-21T20:29:16","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=86488"},"modified":"2026-03-25T12:34:59","modified_gmt":"2026-03-25T17:34:59","slug":"sum-of-interior-angles","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/sum-of-interior-angles\/","title":{"rendered":"Sum of Interior Angles"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_qfyoEzYuk6o\" 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_qfyoEzYuk6o\" data-source-videoID=\"qfyoEzYuk6o\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Sum of Interior Angles Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Sum of Interior Angles\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_qfyoEzYuk6o:hover {cursor:pointer;} img#videoThumbnailImage_qfyoEzYuk6o {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/1731-thumb-final-1.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_qfyoEzYuk6o\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_qfyoEzYuk6o\" 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\",\"Sum of Interior Angles\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_qfyoEzYuk6o\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_qfyoEzYuk6o\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_qfyoEzYuk6o\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction FUX_Function() {\n  var x = document.getElementById(\"FUX\");\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=\"FUX_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=\"FUX\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#What_are_Polygons\" class=\"smooth-scroll\">What are Polygons?<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Triangles\" class=\"smooth-scroll\">Triangles<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Pentagons\" class=\"smooth-scroll\">Pentagons<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Patterns\" class=\"smooth-scroll\">Patterns<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Hexagons\" class=\"smooth-scroll\">Hexagons<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Other_Problems\" class=\"smooth-scroll\">Other Problems<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Sum_of_Interior_Angles_Practice_Questions\" class=\"smooth-scroll\">Sum of Interior Angles 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 the sum of interior angles of polygons! In this video, we will explore different types of polygons and how to find the sum of their interior angles.<\/p>\n<p>Let\u2019s learn about the sum of interior angles!<\/p>\n<h2><span id=\"What_are_Polygons\" class=\"m-toc-anchor\"><\/span>What are Polygons?<\/h2>\n<p>\nPolygons are everywhere we look. The board in the classroom is a rectangle, and the floor tiles are squares. The yield road sign is a triangle, and the stop sign is an octagon.<\/p>\n<p>A polygon is a two-dimensional figure that is formed when line segments meet to create a closed shape. We name polygons based on the number of sides. A triangle has 3 sides, a quadrilateral has 4 sides, and an octagon has 8 sides.<\/p>\n<p>Today we want to learn about how to find the sum of interior angles of any polygon. This is a skill we use in our daily life. When laying tile or even choosing a tile, which will always be shaped like a polygon, we will use the measure of the angles to determine how the tiles will fit with each other.<\/p>\n<h2><span id=\"Triangles\" class=\"m-toc-anchor\"><\/span>Triangles<\/h2>\n<p>\nWe will start with a triangle, since it is the polygon with the least number of sides. To find the sum of the interior angles of this triangle, I am going to rip off all the angles and then put them back together so the vertices meet. <\/p>\n<p>As you can see, when we line up the three angles, they form a straight line, and we know that a straight angle is 180\u00b0. Now let\u2019s look at a polygon with four sides, a rectangle, which is a type of quadrilateral, a polygon with four sides. As you can see, I can use a line segment to connect the opposite angles, and this will turn my rectangle into two triangles.<\/p>\n<p>Since I know the sum of the interior angles of one triangle, we will multiply 180 by two to get the sum of the interior angles of a rectangle, 360\u00b0, or the sum of the interior angles of any quadrilateral. <\/p>\n<h2><span id=\"Pentagons\" class=\"m-toc-anchor\"><\/span>Pentagons<\/h2>\n<p>\nA polygon with 5 sides is called a pentagon. As you can see, I was able to create 3 triangles inside the pentagon. To find the sum of the interior angles of the pentagon, we will multiply 180 by 3, which is 540\u00b0. <\/p>\n<p>As you can imagine, finding the sum of the interior angles of a polygon using this method of creating triangles can become quite tedious if we are dealing with a polygon with 25 sides.<\/p>\n<h2><span id=\"Patterns\" class=\"m-toc-anchor\"><\/span>Patterns<\/h2>\n<p>\nLet us take a closer look and see if we can see a pattern. We will organize the information we have gathered so far into a table.<\/p>\n<table class=\"ATable\" style=\"margin: auto; width: 100%;\">\n<thead>\n<tr style=\"height: 40px;\">\n<th style=\"line-height: 16px; vertical-align: middle;\">Shape<\/th>\n<th style=\"line-height: 16px; vertical-align: middle;\">Number of Sides<\/th>\n<th style=\"line-height: 16px; vertical-align: middle;\">Sum of interior angles<\/th>\n<th style=\"line-height: 16px; vertical-align: middle;\">Pattern<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>Triangle<\/td>\n<td>3<\/td>\n<td>180\u00b0<\/td>\n<td>1 triangle \\(\\times \\) 180\u00b0<\/td>\n<\/tr>\n<tr>\n<td>Quadrilateral<\/td>\n<td>4<\/td>\n<td>360\u00b0<\/td>\n<td>2 triangles \\(\\times \\) 180\u00b0<\/td>\n<\/tr>\n<tr>\n<td>Pentagon<\/td>\n<td>5<\/td>\n<td>540\u00b0<\/td>\n<td>3 triangles \\(\\times \\) 180\u00b0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>\n&nbsp;<br \/>\nThe closer look at the pattern shows that we always multiply two less than the number of sides the polygon has by 180, so we can create a formula, where <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span> represents the number of the sides of the polygon:<\/p>\n<div class=\"examplesentence\">\\((n-2)\\times 180=\\) the sum of the interior angles of a polygon<\/div>\n<p>\n&nbsp;<\/p>\n<h2><span id=\"Hexagons\" class=\"m-toc-anchor\"><\/span>Hexagons<\/h2>\n<p>\nLet\u2019s test our formula:<\/p>\n<p>What is the sum of the interior angles of a hexagon?<\/p>\n<p>We know that a hexagon has 6 sides. Therefore, <span style=\"font-style:normal; font-size:90%\">\\((6 \u2013 2)\\times 180 = 720\u00b0\\)<\/span>, which is the sum of the interior angles of a hexagon.<\/p>\n<p>Let us confirm our finding by drawing a hexagon and dividing it into triangles. As you can see, there are 4 triangles and  <span style=\"font-style:normal; font-size:90%\">\\(4\\times 180 = 720\\)<\/span>.<\/p>\n<h2><span id=\"Other_Problems\" class=\"m-toc-anchor\"><\/span>Other Problems<\/h2>\n<p>\nNow that we have a formula, we can solve many more types of problems. <\/p>\n<p>Here\u2019s another example:<\/p>\n<p>The sum of the interior angles of a polygon is 1,080\u00b0. How many sides does the polygon have?<\/p>\n<p>We can use the formula and substitute what we have <span style=\"font-style:normal; font-size:90%\">\\((n-2)\\times 180=1,080\\)<\/span>. Since <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span> represents the number of sides, we can use our algebra skills to solve for <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span>. First, we divide both sides by 180, which will result in, <span style=\"font-style:normal; font-size:90%\">\\(n-2=6\\)<\/span>. Then we add 2 to both sides to get <span style=\"font-style:normal; font-size:90%\">\\(n=8\\)<\/span>. Therefore, the polygon with interior angles adding to 1,080\u00b0 is an octagon. <\/p>\n<p>Now that we\u2019ve learned how to find the sum of the interior angles of a polygon, let\u2019s try to find the measure of one angle of a regular polygon. As you recall, a regular polygon is equiangular and equilateral, which means all the angles have the same measure and the sides have the same length. Since the formula, <span style=\"font-style:normal; font-size:90%\">\\((n-2)\\times 180\\)<\/span>, gives us the sum of the interior angles, if we divide by the total by the number of angles, it will give us the measure of one angle. Therefore:<\/p>\n<div class=\"examplesentence\">\\(\\frac{(n-2)180}{n}=\\) one interior angle of a regular polygon<\/div>\n<p>\n&nbsp;<br \/>\nLet\u2019s take a look at an example.<\/p>\n<p>The measure of one interior angle of a regular polygon is 150\u00b0. How many sides does the polygon have?<\/p>\n<p>We will start by substituting what we know into the formula, <span style=\"font-style:normal; font-size:90%\">\\(\\frac{(n-2)180}{n}=150\\)<\/span>. We will eliminate the fraction by multiplying both sides by <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span> to get <span style=\"font-style:normal; font-size:90%\">\\((n-2)180=150n\\)<\/span>. Simplify using the distributive property, <span style=\"font-style:normal; font-size:90%\">\\(180n \u2013 360 = 150n\\)<\/span>. Solve for <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span> by combining like terms, <span style=\"font-style:normal; font-size:90%\">\\(30n=360\\)<\/span>. Then isolate the variable by dividing both sides by 30 to get <span style=\"font-style:normal; font-size:90%\">\\(n=12\\)<\/span>. Therefore, the regular polygon, where the measure of one interior angle is 150\u00b0 has 12 sides, which is called a dodecagon.<\/p>\n<p>Now let\u2019s talk about the exterior angles. When we extend the sides of a polygon, the angle created on the outside is called an exterior angle. As you can see, the exterior angle and the interior angle form a straight angle.<\/p>\n<p>Now, we will consider this regular triangle. This means that the measure of each interior angle is 60\u00b0. If the interior angle is 60\u00b0, then the exterior angle must be 120\u00b0. We can add the measure of the 3 exterior angles for this triangle to find the sum of the exterior angles of a triangle, which is 360\u00b0.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89923\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-1.png\" alt=\"Image of a triangle with angles of 60 degrees, and exterior angles of 120 degrees.\" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-1.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-1-300x169.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-1-1024x576.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-1-768x432.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-1-1536x864.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>Now let\u2019s take a look at a rectangle. We will do the same thing and extend each side. We know that the interior angles are 90\u00b0, which makes the exterior angles also 90\u00b0. We will add the exterior angles to find the sum of the exterior angles of a rectangle, which is <span style=\"font-style:normal; font-size:90%\">\\(90\\times 4=360\u00b0\\)<\/span>. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-89926\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-2.png\" alt=\"Image of rectangular polygon with angles of 90 degrees, and exterior angles of 90 degrees. \" width=\"777\" height=\"437\"style=\"box-shadow: 1.5px 1.5px 3px grey\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-2.png 1920w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-2-300x168.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-2-1024x574.png 1024w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-2-768x431.png 768w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/08\/Sum-of-Interior-Angles-2-1536x862.png 1536w\" sizes=\"auto, (max-width: 777px) 100vw, 777px\" \/><\/p>\n<p>Based on our two experiments, we can conclude that the sum of the exterior angles of any polygon is always 360\u00b0. This knowledge can also help us solve problems. <\/p>\n<p>For example, the measure of one exterior angle of a regular polygon is 72\u00b0. How many sides does this regular polygon have? <\/p>\n<p>Since this is a regular polygon and we know that the sum of the exterior angles of all polygons is 360\u00b0, we can simply create the equation <span style=\"font-style:normal; font-size:90%\">\\(\\frac{360}{n}=72\\)<\/span>. Then we solve for <span style=\"font-style:normal; font-size:90%\">\\(n\\)<\/span> to find the number of sides. We will first eliminate the fraction, <span style=\"font-style:normal; font-size:90%\">\\(72n=360\\)<\/span>. Then divide both sides by 72 to isolate the variable, which gives us that <span style=\"font-style:normal; font-size:90%\">\\(n=5\\)<\/span>. Therefore, the polygon has 5 sides, which is also called a pentagon.<\/p>\n<p>I hope you enjoyed this video about the sum of the interior angles of polygons. Thanks for watching, and happy studying!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Sum_of_Interior_Angles_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Sum of Interior Angles 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 \/>\nWhat is the sum of interior angles of any nonagon, or nine-sided polygon?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-1-1\">940\u00b0<\/div><div class=\"PQ\"  id=\"PQ-1-2\">1,080\u00b0<\/div><div class=\"PQ correct_answer\"  id=\"PQ-1-3\">1,260\u00b0<\/div><div class=\"PQ\"  id=\"PQ-1-4\">1,350\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>In order to find the sum of the interior angles of a polygon, use the formula \\((n-2)\\times180\\), where \\(n\\) is the number of sides of the polygon.<\/p>\n<p>In this case, the polygon of concern has nine sides, so the sum of interior angles is equal to \\((9-2)\\times180\\), or \\(7\\times180\\), which equals 1,260 degrees.<\/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 \/>\nWhat polygon has a sum of interior angles equal to 2,160\u00b0?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\">A thirteen-sided polygon<\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\">A fourteen-sided polygon<\/div><div class=\"PQ\"  id=\"PQ-2-3\">A fifteen-sided polygon<\/div><div class=\"PQ\"  id=\"PQ-2-4\">A sixteen-sided polygon<\/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>Remember that the sum of interior angles of a polygon is equal to \\((n-2)\\times180\\). Set this equal to 2,160 and then solve for \\(n\\) to determine how many sides the polygon has.<\/p>\n<p style=\"text-align:center;\">\\((n-2)\\times180=2{,}160\\)<\/p>\n<p>Divide both sides by 180.<\/p>\n<p style=\"text-align:center; line-height: 55px;\">\n\\((n-2)=\\dfrac{2{,}160}{180}\\)<br \/>\n\\(n-2=12\\)<\/p>\n<p>Finally, add 2 to both sides.<\/p>\n<p style=\"text-align:center;\">\\(n=14\\)<\/p>\n<p>So fourteen-sided polygons have a sum of interior angles equal to 2,160\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 \/>\nHow many degrees are in one interior angle of a regular decagon (ten-sided polygon)?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-3-1\">144\u00b0<\/div><div class=\"PQ\"  id=\"PQ-3-2\">152\u00b0<\/div><div class=\"PQ\"  id=\"PQ-3-3\">156\u00b0<\/div><div class=\"PQ\"  id=\"PQ-3-4\">160\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>Since the interior angles of regular polygons are all equal in measure, the measure of one of the angles can be found by dividing the sum of interior angles by \\(n\\).<\/p>\n<p style=\"text-align:center;\">\\(\\dfrac{(n-2)\\times180}{n}\\)<\/p>\n<p>The problem states that this polygon has ten sides, so plug \\(n=10\\) into the formula.<\/p>\n<p style=\"text-align:center; line-height: 50px\">\\(\\dfrac{(10-2)\\times180}{10}=\\dfrac{8\\times180}{10}\\)\\(\\:=144\u00b0\\)<\/p>\n<p>Each interior angle of a regular decagon has measure equal to 144\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 \/>\nChalice is designing a playground merry-go-round in the shape of a regular polygon, of which each interior angle is equal to 135\u00b0. How many sides does the merry-go-round have?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">7<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-2\">8<\/div><div class=\"PQ\"  id=\"PQ-4-3\">9<\/div><div class=\"PQ\"  id=\"PQ-4-4\">11<\/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>Since the measure of one interior angle of a regular polygon can be found with the following formula, the number of sides can be determined by solving for \\(n\\):<\/p>\n<p style=\"text-align: center\">\\(\\dfrac{(n-2)\\times180}{n}\\)<\/p>\n<p>First, set this formula equal to 135\u00b0.<\/p>\n<p style=\"text-align:center;\">\\(\\dfrac{(n-2)\\times180}{n}=135\\)<\/p>\n<p>Multiply both sides by \\(n\\).<\/p>\n<p style=\"text-align:center;\">\\((n-2)\\times180=135n\\)<\/p>\n<p>Multiply out the left side to get:<\/p>\n<p style=\"text-align:center;\">\\(180n-360=135n\\)<\/p>\n<p>Then add 360 and subtract 135\\(n\\) from both sides.<\/p>\n<p style=\"text-align:center; line-height: 35px;\">\n\\(180n=135n+360\\)<br \/>\n\\(45n=360\\)<\/p>\n<p>Divide both sides by 45.<\/p>\n<p style=\"text-align:center;\">\\(n=8\\)<\/p>\n<p>Therefore, the merry-go-round has eight sides.<\/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 \/>\nReagan is designing some table legs in a wood shop. Each leg is shaped with an equal number of sides such that the exterior angles are all equal to 24\u00b0. How many sides does each leg have?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">13<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-2\">15<\/div><div class=\"PQ\"  id=\"PQ-5-3\">17<\/div><div class=\"PQ\"  id=\"PQ-5-4\">19<\/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 exterior angles of a polygon must add up to 360\u00b0. Since this problem concerns regular polygons, each angle is equal in measure. Because of this, you can use the formula \\(\\frac{360}{n}=24\u00b0\\) to solve for the number of sides.<\/p>\n<p>Multiply both sides by \\(n\\), then divide both sides by 24.<\/p>\n<p style=\"text-align:center; line-height: 50px;\">\n\\(360=24n\\)<br \/>\n\\(\\dfrac{360}{24}=n\\)<br \/>\n\\(15=n\\)<\/p>\n<p>This means the table legs each have 15 sides.<\/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":100732,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-86488","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-angle-videos","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\/86488","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=86488"}],"version-history":[{"count":6,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/86488\/revisions"}],"predecessor-version":[{"id":281825,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/86488\/revisions\/281825"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/100732"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=86488"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}