{"id":52625,"date":"2019-08-13T14:56:05","date_gmt":"2019-08-13T14:56:05","guid":{"rendered":"https:\/\/www.mometrix.com\/academy\/?page_id=52625"},"modified":"2026-03-25T11:04:58","modified_gmt":"2026-03-25T16:04:58","slug":"polygons","status":"publish","type":"page","link":"https:\/\/www.mometrix.com\/academy\/polygons\/","title":{"rendered":"Polygons"},"content":{"rendered":"\n\t\t\t<div id=\"mmDeferVideoEncompass_z1g4DNNXsuI\" 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_z1g4DNNXsuI\" data-source-videoID=\"z1g4DNNXsuI\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/01\/circle-play-duotone.png\" alt=\"Polygons Video\" height=\"464\" width=\"825\" class=\"size-full\" data-matomo-title = \"Polygons\">\n\t\t\t<\/picture>\n\t\t\t<\/div>\n\t\t\t<style>img#videoThumbnailImage_z1g4DNNXsuI:hover {cursor:pointer;} img#videoThumbnailImage_z1g4DNNXsuI {background-size:contain;background-image:url(\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2023\/07\/updated-polygons-64bec6cbc4209.webp\");}<\/style>\n\t\t\t<script defer>\n\t\t\t  jQuery(\"img#videoThumbnailImage_z1g4DNNXsuI\").click(function() {\n\t\t\t\tlet videoId = jQuery(this).attr(\"data-source-videoID\");\n\t\t\t\tlet helpTag = '<div id=\"mmDeferVideoYTMessage_z1g4DNNXsuI\" 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\",\"Polygons\");\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_z1g4DNNXsuI\").html(tag);\n\t\t\t\tjQuery(\"div#mmDeferVideoEncompass_z1g4DNNXsuI\").prepend(helpTag);\n\t\t\t\tsetTimeout(function(){jQuery(\"div#mmDeferVideoYTMessage_z1g4DNNXsuI\").css(\"display\", \"block\");}, 2000);\n\t\t\t  });\n\t\t\t  \n\t\t\t<\/script>\n\t\t\n<p><script>\nfunction iQI_Function() {\n  var x = document.getElementById(\"iQI\");\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=\"iQI_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=\"iQI\" style=\"display:none;\">\n<ul>\n<li class=\"toc-h2\"><a href=\"#How_Polygons_are_Constructed\" class=\"smooth-scroll\">How Polygons are Constructed<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Ngon\" class=\"smooth-scroll\">N-gon<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Regular_vs_Irregular\" class=\"smooth-scroll\">Regular vs. Irregular<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Convex_vs_Concave\" class=\"smooth-scroll\">Convex vs. Concave<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Finding_Diagonals\" class=\"smooth-scroll\">Finding Diagonals<\/a><\/li>\n<li class=\"toc-h2\"><a href=\"#Polygon_Practice_Questions\" class=\"smooth-scroll\">Polygon 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>Hi and welcome to this video about polygons! In this video, we will explore four things:<\/p>\n<ol>\n<li>What polygons are<\/li>\n<li>The different parts of a polygon<\/li>\n<li>Ways to classify polygons <\/li>\n<li>How to determine the number of diagonals in a polygon<\/li>\n<\/ol>\n<p>The term <strong>polygon<\/strong> is derived from the Greek words <em>polys<\/em> meaning \u201cmany\u201d and <em>gonia<\/em> meaning \u201cangle\u201d. So, polygons have many <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/angles\/\">angles<\/a>.<\/p>\n<h2><span id=\"How_Polygons_are_Constructed\" class=\"m-toc-anchor\"><\/span>How Polygons are Constructed<\/h2>\n<p>\nFirst let\u2019s explore how polygons are constructed.<\/p>\n<h3><span id=\"Zero_Dimensions\" class=\"m-toc-anchor\"><\/span>Zero Dimensions<\/h3>\n<p>\nConsider the geometric point, represented by a dot:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65213\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/dot-300x201.png\" alt=\"dot on board\" width=\"300\" height=\"201\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/dot-300x201.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/dot.png 640w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>The point is 0-dimensional\u2014it has no length, width, height, nothing.<\/p>\n<h3><span id=\"One_Dimension\" class=\"m-toc-anchor\"><\/span>One Dimension<\/h3>\n<p>\nNow, let\u2019s consider two connected points: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65227\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/line-segment-300x154.png\" alt=\"line segment\" width=\"300\" height=\"154\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/line-segment-300x154.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/line-segment.png 704w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>This is called a <a class=\"ylist\" href=\"https:\/\/www.mometrix.com\/academy\/lines-and-planes\/\">line<\/a>, or a line segment. A line segment is 1-dimensional. It has length, but no width or height.<\/p>\n<h3><span id=\"Two_Dimensions\" class=\"m-toc-anchor\"><\/span>Two Dimensions<\/h3>\n<p>\nWhen multiple line segments are connected end-to-end, polygons such as this, can be formed:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65231\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/triangle-300x146.png\" alt=\"triangle\" width=\"300\" height=\"146\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/triangle-300x146.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/triangle.png 697w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Polygons are two-dimensional. They have no thickness, like this.<\/p>\n<p>In order to be a polygon, the shape must be <em>closed<\/em>. In other words, every endpoint must be connected to another endpoint. This figure is comprised of connected segments, but the result is not a polygon.<\/p>\n<p>This shape is closed, but not made up of connected segments, so it is also not a polygon:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65219\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/examples-191x300.png\" alt=\"examples of polygons and non-polygons\" width=\"191\" height=\"300\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/examples-191x300.png 191w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/examples.png 453w\" sizes=\"auto, (max-width: 191px) 100vw, 191px\" \/><\/p>\n<h3><span id=\"Edges_and_Vertices\" class=\"m-toc-anchor\"><\/span>Edges and Vertices<\/h3>\n<p>\nThe sides of polygons are called <strong>edges<\/strong> and the angles created where the edges intersect are called <strong>vertices<\/strong>. Polygons also have as many edges as vertices:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65216\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/edges-and-vertices-300x158.png\" alt=\"triangle edges and vertices\" width=\"300\" height=\"158\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/edges-and-vertices-300x158.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/edges-and-vertices.png 584w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Polygons are named by the number of edges they have. This polygon has three edges and three vertices and is called a triangle. The triangle has the smallest number of edges and vertices of any polygon\u2014it is impossible to create a two-sided polygon.<\/p>\n<p>Some common polygons are quadrilaterals (which have four sides), pentagons (which have five sides), hexagons (six sides), heptagons (seven sides), octagons (eight sides), nonagons (nine sides), decagons (ten sides) and dodecagons (which have twelve sides). <\/p>\n<h2><span id=\"Ngon\" class=\"m-toc-anchor\"><\/span>N-gon<\/h2>\n<p>\nThough polygons with any number of edges have names, the general \\(n\\)-gon is typically used for all other polygons, where \\(n\\) represents the number of sides. For instance, a 30-sided polygon is called a triacontagon, but it\u2019s often simply called a 30-gon.<\/p>\n<h2><span id=\"Regular_vs_Irregular\" class=\"m-toc-anchor\"><\/span>Regular vs. Irregular<\/h2>\n<p>\nPolygons can be regular or irregular. <strong>Regular polygons<\/strong> have congruent edges and congruent vertices. For example:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65230\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/quadrilateral-300x182.png\" alt=\"quadrilateral\" width=\"300\" height=\"182\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/quadrilateral-300x182.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/quadrilateral.png 497w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>This is a regular quadrilateral. The edges are the same length and the vertices have the same measure.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65224\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/irregular-quadrilaterals-228x300.png\" alt=\"irregular quadrilaterals\" width=\"228\" height=\"300\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/irregular-quadrilaterals-228x300.png 228w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/irregular-quadrilaterals.png 395w\" sizes=\"auto, (max-width: 228px) 100vw, 228px\" \/><\/p>\n<p>This is an <strong>irregular quadrilateral<\/strong>. The vertices have the same measure, but the edges have different lengths.<\/p>\n<p>This is an irregular quadrilateral. The edges are the same length, but the vertices have different measures.<\/p>\n<p>This is an irregular quadrilateral. Neither the edges nor the vertices have the same measure.<\/p>\n<h2><span id=\"Convex_vs_Concave\" class=\"m-toc-anchor\"><\/span>Convex vs. Concave<\/h2>\n<p>\nPolygons can also be convex or concave. When one or more vertices of a polygon measures more than 180 degrees, the result is a <strong>concave polygon<\/strong>. For example: <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65212\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/convex-polygon-300x175.png\" alt=\"hexagon\" width=\"300\" height=\"175\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/convex-polygon-300x175.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/convex-polygon.png 421w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>This is a <strong>convex<\/strong> hexagon. All of the vertices measure less than 180 degrees.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65209\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/concave-300x156.png\" alt=\"concave polygon\" width=\"300\" height=\"156\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/concave-300x156.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/concave.png 411w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>This is a concave hexagon. One vertex measures more than 180\u00b0.<\/p>\n<p>Now remember:<\/p>\n<ul>\n<li>Only polygons with 4 or more sides can be concave because it\u2019s not possible for a triangle to contain an angle measuring more than 180\u00b0, which is a straight line.<\/li>\n<li>Concave polygons cannot be regular because all the vertices will never be the same measure.<\/li>\n<\/ul>\n<p>Polygons also contain diagonals. <strong>Diagonals<\/strong> are line segments joining two vertices that are not next to each other. <\/p>\n<p>As you can see here, this irregular convex pentagon has five diagonals.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65200\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/5-diagonals-300x172.png\" alt=\"pentagon diagonals\" width=\"300\" height=\"172\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/5-diagonals-300x172.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/5-diagonals.png 376w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>This is an irregular concave pentagon.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65206\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/concave-diagonals-300x157.png\" alt=\"concave diagonals\" width=\"300\" height=\"157\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/concave-diagonals-300x157.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/concave-diagonals.png 416w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p> It also has five diagonals, even though the concavity causes diagonals to lie outside the polygon.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-medium wp-image-65203\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/concave-diagonals-drawn-300x161.png\" alt=\"concave diagonals drawn\" width=\"300\" height=\"161\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/concave-diagonals-drawn-300x161.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/concave-diagonals-drawn.png 393w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Triangles do not have diagonals because there is no way to connect two vertices with segments that are not edges.<\/p>\n<p>Now that you know the basics of polygons, let\u2019s use the diagrams to figure out a formula for finding the number of diagonals in any polygon.<\/p>\n<h2><span id=\"Finding_Diagonals\" class=\"m-toc-anchor\"><\/span>Finding Diagonals<\/h2>\n<p>\nIn this case, the number of diagonals connecting to each vertex is 2, which is 3 less than the number of vertices, 5: 2 diagonals connected to each vertex.<\/p>\n<p>This is true of all diagonals of all polygons. A nonagon has \\(9-3=6\\) diagonals connecting to each vertex. <\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-65228\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/01\/nonagon.png\" alt=\"nonagon\" width=\"297\" height=\"257\" \/><\/p>\n<p>The number 3 is not arbitrary here \u2013 from any vertex, diagonals cannot connect to the vertex itself or to the vertices that they are \u201c1 away\u201d from, because it would be edges. That makes 3 vertices from every vertex that aren\u2019t included. To generalize this, we\u2019ll use \\(n-3\\), where <em>n<\/em> is the number of vertices of the polygon. <\/p>\n<p>Now, each vertex has the same number of diagonals connecting to it, so in this case, we can see that the total number of diagonal connections to vertices is \\(5(5-3)=10\\).<\/p>\n<p>In general, we can say the total number of diagonal connections is \\(n(n-3)\\).<\/p>\n<p>When we figure out this total, though, we are counting each diagonal twice, because diagonals have 2 endpoints. In order to figure out the number of unique diagonals, we need to divide our total by 2. In our pentagon, this looks like: <\/p>\n<div class=\"examplesentence\">\\(\\frac{5(5-3)}{2} = 5\\) unique diagonals<\/div>\n<p>\n&nbsp;<br \/>\nTherefore, for any sized polygon, our equation can be written as this:<\/p>\n<div class=\"examplesentence\">\\(\\frac{n(n-3)}{2}\\)<\/div>\n<p>\n&nbsp;<br \/>\nConceptually, this can be remembered as:<\/p>\n<div class=\"examplesentence\">\\(\\frac{(\\text{total number of vertices})(\\text{number of diagonal connections at each vertex})}{(\\text{number of endpoints of each diagonal})}\\)<\/div>\n<p>\n&nbsp;<br \/>\nUsing our formula, we can determine the number of unique diagonals in, for example, a 17-gon:<\/p>\n<div class=\"examplesentence\">\\(\\frac{17(17-3)}{2} = 119\\text{ unique diagonals}\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe can also see algebraically that triangles have no diagonals:<\/p>\n<div class=\"examplesentence\">\\(\\frac{3(3-3)}{2} = 0\\)<\/div>\n<p>\n&nbsp;<br \/>\nWe can also figure out how many edges or vertices a polygon has by the number of unique diagonals.<\/p>\n<p>Suppose a polygon has 44 unique diagonals. How many edges does the polygon have?<\/p>\n<div class=\"examplesentence\">\\(\\frac{n(n-3)}{2} = 44\\)<br \/>\n\\(n(n-3) = 88\\)<br \/>\n\\(n^2 &#8211; 3n = 88\\)<br \/>\n\\(n^2 &#8211; 3n &#8211; 88 = 0\\)<br \/>\n\\((n + 8)(n-11) = 0\\)<br \/>\n\\(n = -8\\text{ or }n = 11\\)\n<\/div>\n<p>\n&nbsp;<br \/>\nAlright, so here we are at our answer, \\(n\\) (the number of edges the polygon has) is either equal to 11 or -8. Because what we want to do is we want to make sure that we end with a result of 0, which means that \\(n\\), somewhere, has to be 11 or -8 in order to get us multiplying 0 by another answer, resulting in 0. Since \\(n\\) can\u2019t be negative (cause we can\u2019t have negative edges), it can\u2019t be -8. So we know that our answer is 11. The number of edges is 11, making it an 11-gon.<\/p>\n<p>Alright, excellent! I hope this video helped increase your knowledge of polygons and made some of their properties a little bit clearer! Thanks for watching, see you next time!<\/p>\n<\/div>\n<div class=\"spoiler\" id=\"PQs-spoiler\">\n<h2 style=\"text-align:center\"><span id=\"Polygon_Practice_Questions\" class=\"m-toc-anchor\"><\/span>Polygon 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>Which of the following is a polygon?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-1-1\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-69887 alignnone\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/closed-image-with-lots-of-sides.png\" alt=\"closed image with lots of sides\" width=\"256\" height=\"207\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/closed-image-with-lots-of-sides.png 834w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/closed-image-with-lots-of-sides-300x242.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/closed-image-with-lots-of-sides-768x621.png 768w\" sizes=\"auto, (max-width: 256px) 100vw, 256px\" \/><\/div><div class=\"PQ\"  id=\"PQ-1-2\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-69884 alignnone\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/circle.png\" alt=\"circle\" width=\"256\" height=\"235\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/circle.png 730w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/circle-300x275.png 300w\" sizes=\"auto, (max-width: 256px) 100vw, 256px\" \/><\/div><div class=\"PQ\"  id=\"PQ-1-3\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-69908 alignnone\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/swirly-line.png\" alt=\"swirly line\" width=\"256\" height=\"223\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/swirly-line.png 766w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/swirly-line-300x262.png 300w\" sizes=\"auto, (max-width: 256px) 100vw, 256px\" \/><\/div><div class=\"PQ\"  id=\"PQ-1-4\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-69881 alignnone\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/circle-with-slash-through-it.png\" alt=\"circle with slash through it\" width=\"217\" height=\"218\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/circle-with-slash-through-it.png 641w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/circle-with-slash-through-it-300x300.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/circle-with-slash-through-it-150x150.png 150w\" sizes=\"auto, (max-width: 217px) 100vw, 217px\" \/><\/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>This shape is closed (every endpoint is connected to another endpoint), which is only possible when there are three or more &#8220;endpoints&#8221; (vertices). There are just as many sides as there are vertices, making this a true polygon.<\/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>Which of the following is an irregular polygon?<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-2-1\"><img decoding=\"async\" class=\" wp-image-69911 alignnone\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/triangle.png\" alt=\"triangle\" width=\"217.6\" height=\"192.95\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/triangle.png 849w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/triangle-300x266.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/triangle-768x681.png 768w\" sizes=\"(max-width: 849px) 100vw, 849px\" \/><\/div><div class=\"PQ correct_answer\"  id=\"PQ-2-2\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-69902 alignnone\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/right-triangle.png\" alt=\"right triangle\" width=\"128\" height=\"206\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/right-triangle.png 476w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/right-triangle-186x300.png 186w\" sizes=\"auto, (max-width: 128px) 100vw, 128px\" \/><\/div><div class=\"PQ\"  id=\"PQ-2-3\"><img decoding=\"async\" class=\" wp-image-69905 alignnone\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/square-.png\" alt=\"square\" width=\"177.65\" height=\"167.45\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/square-.png 801w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/square--300x283.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/square--768x725.png 768w\" sizes=\"(max-width: 801px) 100vw, 801px\" \/><\/div><div class=\"PQ\"  id=\"PQ-2-4\"><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-69893 alignnone\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/hexagon.png\" alt=\"hexagon\" width=\"171\" height=\"157\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/hexagon.png 778w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/hexagon-300x277.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/hexagon-768x709.png 768w\" sizes=\"auto, (max-width: 171px) 100vw, 171px\" \/><\/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>While the equilateral triangle (Shape A), the square (Shape C), and the regular hexagon (Shape D) each have edges that are each the same length and same-degree angles at each vertex, the right triangle shown in Choice B has edges and angles that measure different lengths\/degrees.<\/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>Is this a concave or a convex polygon?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-69899 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/polygon-with-a-side-caving-in.png\" alt=\"polygon with a side caving in\" width=\"256\" height=\"236\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/polygon-with-a-side-caving-in.png 779w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/polygon-with-a-side-caving-in-300x277.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/polygon-with-a-side-caving-in-768x708.png 768w\" sizes=\"auto, (max-width: 256px) 100vw, 256px\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ correct_answer\"  id=\"PQ-3-1\">Concave<\/div><div class=\"PQ\"  id=\"PQ-3-2\">Convex<\/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>This is a concave polygon, and we can see this in the top, right-hand portion of the shape:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-69890 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/concave-polygon.png\" alt=\"concave polygon\" width=\"256\" height=\"236\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/concave-polygon.png 778w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/concave-polygon-300x277.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/concave-polygon-768x709.png 768w\" sizes=\"auto, (max-width: 256px) 100vw, 256px\" \/><\/p>\n<p>This angle sitting inside of the polygon is larger than 180\u00b0. There\u2019s a portion of the shape &#8220;caving in on itself.&#8221;<\/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>How many unique diagonals does this polygon have?<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\" wp-image-69896 aligncenter\" src=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/octogon.png\" alt=\"octogon\" width=\"256\" height=\"237\" srcset=\"https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/octogon.png 778w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/octogon-300x277.png 300w, https:\/\/www.mometrix.com\/academy\/wp-content\/uploads\/2021\/03\/octogon-768x710.png 768w\" sizes=\"auto, (max-width: 256px) 100vw, 256px\" \/><\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-4-1\">48<\/div><div class=\"PQ\"  id=\"PQ-4-2\">40<\/div><div class=\"PQ\"  id=\"PQ-4-3\">28<\/div><div class=\"PQ correct_answer\"  id=\"PQ-4-4\">20<\/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 this shape is a relatively simple shape, we could feasibly count the diagonals by hand, but it would be more efficient to use our given formula for determining the amount of unique diagonals in any polygon:<\/p>\n<p style=\"text-align: center\">\\(\\dfrac{n(n-3)}{2}\\)<\/p>\n<p>Here, \\(n\\) represents the amount of vertices that that polygon has.<\/p>\n<p>This octagon has eight vertices, so we find that \\(\\frac{8(8-3)}{2}=20\\).<\/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>Given that some \\(n\\)-gon has 77 unique diagonals, find the value of \\(n\\).<\/p>\n<\/div>\n\t\t\t\t\t<div class=\"PQ-Choices\"><div class=\"PQ\"  id=\"PQ-5-1\">11<\/div><div class=\"PQ\"  id=\"PQ-5-2\">10<\/div><div class=\"PQ correct_answer\"  id=\"PQ-5-3\">14<\/div><div class=\"PQ\"  id=\"PQ-5-4\">7<\/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>This time, we must work backwards with our formula. We set our equation up like this:<\/p>\n<p style=\"text-align:center;\">\n\\(\\dfrac{n(n-3)}{2}=77\\)<\/p>\n<p>Our goal is to solve for \\(n\\).<\/p>\n<p style=\"text-align:center; line-height: 50px\">\n\\(\\frac{n(n-3)}{2}=77\\)<br \/>\n\\(n(n-3)=154\\)<br \/>\n\\(n^2-3n=154\\)<br \/>\n\\(n^2-3n-154=0\\)<\/p>\n<p>At this point, we\u2019ll need to employ the quadratic formula to find the two possible solutions\u2026<\/p>\n<p style=\"text-align:center; line-height: 45px;\">\n\\(n=\\frac{-(-3)\\pm\\sqrt{(-3)^2-4\\times(1)\\times(154)}}{2\\times1}\\)<br \/>\n\\(n=\\frac{3\\pm\\sqrt{9+(4\\times154)}}{2}\\)<br \/>\n\\(n=\\frac{3\u00b1\\sqrt{9+616}}{2}\\)<br \/>\n\\(n=\\frac{3\u00b1\\sqrt{625}}{2}\\)<br \/>\n\\(n=\\frac{3\u00b125}{2}\\)<br \/>\n\\(n=14\\) or \\(n=-11\\)<\/p>\n<p>Out of these two possible solutions for \\(n\\), only one of them makes sense for us! The value of \\(n\\) must be \\(n=14\\) because we cannot have a negative number of edges or vertices in a polygon.<\/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":185942,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":{"0":"post-52625","1":"page","2":"type-page","3":"status-publish","4":"has-post-thumbnail","6":"page_category-math-advertising-group","7":"page_category-shape-videos","8":"page_type-video","9":"content_type-practice-questions","10":"subject_matter-math"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/52625","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=52625"}],"version-history":[{"count":5,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/52625\/revisions"}],"predecessor-version":[{"id":279121,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/pages\/52625\/revisions\/279121"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media\/185942"}],"wp:attachment":[{"href":"https:\/\/www.mometrix.com\/academy\/wp-json\/wp\/v2\/media?parent=52625"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}