Inscribed/Circumscribed Polygons
<p><span style="font-size:22px"><a href="https://www.prepswift.com/quizzes/quiz/prepswift-inscribedcircumscribed-polygons" target="_blank">Inscribed/Circumscribed Polygons Exercise</a></span></p>
<p><strong>Note</strong>: about 2.5 minutes into the video, Greg describes a case where an "octagon" does not inscribe a circle. The problem is that it isn't actually an octagon - that polygon has 10 sides (also called as a decagon).</p><p>If you read the (optional) Archimedes Calculates $\pi$, you already have a sense of <strong><span style="color:#8e44ad;">Inscribed and Circumscribed Polygons</span></strong>. </p>
<p style="margin-left: 40px;"><strong><span style="color:#27ae60;">Inscribed Polygon</span></strong>: A polygon that is inside a circle where all of the polygon's vertices sit on the circumference of the circle.</p>
<center><svg height="218.77500915527344" style="
width:661.6000366210938px;
height:218.77500915527344px;
background: #FFF;
fill: none;
" width="661.6000366210938" x="0" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" y="0"> <svg class="role-diagram-draw-area" xmlns="http://www.w3.org/2000/svg"><g class="shapes-region" style="stroke: black; fill: none;"><g class="composite-shape"><path class="real" d=" M69,89.3 C69,48.04 102.44,14.6 143.7,14.6 C184.96,14.6 218.4,48.04 218.4,89.3 C218.4,130.56 184.96,164 143.7,164 C102.44,164 69,130.56 69,89.3 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M259,88.3 C259,47.04 292.44,13.6 333.7,13.6 C374.96,13.6 408.4,47.04 408.4,88.3 C408.4,129.56 374.96,163 333.7,163 C292.44,163 259,129.56 259,88.3 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M259.74,86.62 L335.38,14.34 L407.66,89.98 L332.02,162.26 Z" style="stroke-width: 1; stroke: rgb(208, 2, 27); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M143.7,15.6 L215.2,109.3 L72.2,109.3 Z" style="stroke-width: 1; stroke: rgb(74, 144, 226); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M452,87.3 C452,46.04 485.44,12.6 526.7,12.6 C567.96,12.6 601.4,46.04 601.4,87.3 C601.4,128.56 567.96,162 526.7,162 C485.44,162 452,128.56 452,87.3 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M601.4,87.3 L579.52,140.12 L526.7,162 L473.88,140.12 L452,87.3 L473.88,34.48 L526.7,12.6 L579.52,34.48 Z" style="stroke-width: 1; stroke: rgb(189, 16, 224); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g></g></g><g></g><g></g><g></g></svg> <svg height="217.1750030517578" style="width:660px;height:217.1750030517578px;font-family:Asana-Math, Asana;background:#FFF;" width="660" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:13.5px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="86.80001831054688" y="176.40000915527344">triangle inscribed</text></g></g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:13.5px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="114.94375610351562" y="195.3000030517578">in circle</text></g></g></g><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:13.5px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="282.8000183105469" y="176.40000915527344">square inscribed</text></g></g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:13.5px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="309.4437561035156" y="195.3000030517578">in circle</text></g></g></g><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:13.5px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="478.8000183105469" y="176.40000915527344">octagon inscribed</text></g></g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:13.5px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="508.8249816894531" y="195.3000030517578">in circle</text></g></g></g></svg> </svg></center>
<p style="margin-left: 40px;"><strong><span style="color:#27ae60;">Circumscribed Polygon</span></strong>: A polygon surrounding a circle so that each of its sides touches the circumference of the circle.</p>
<center><svg height="188.01499938964844" style="
width:661.9199829101562px;
height:188.01499938964844px;
background: #FFF;
fill: none;
" width="661.9199829101562" x="0" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" y="0"> <svg class="role-diagram-draw-area" xmlns="http://www.w3.org/2000/svg"><g class="shapes-region" style="stroke: black; fill: none;"><g class="composite-shape"><path class="real" d=" M192.35,7.8 L285.2,133.15 L98.2,133.15 Z" style="stroke-width: 1; stroke: rgb(74, 144, 226); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M145.87,85.96 C145.87,60.47 166.54,39.8 192.04,39.8 C217.53,39.8 238.2,60.47 238.2,85.96 C238.2,111.46 217.53,132.12 192.04,132.12 C166.54,132.12 145.87,111.46 145.87,85.96 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M490.2,66.92 L453.16,131.08 L379.08,131.08 L342.04,66.92 L379.08,2.76 L453.16,2.76 Z" style="stroke-width: 1; stroke: rgb(74, 144, 226); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M352.82,66.92 C352.82,31.96 381.16,3.62 416.12,3.62 C451.08,3.62 479.42,31.96 479.42,66.92 C479.42,101.88 451.08,130.22 416.12,130.22 C381.16,130.22 352.82,101.88 352.82,66.92 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g></g></g><g></g><g></g><g></g></svg> <svg height="186.09500122070312" style="width:660px;height:186.09500122070312px;font-family:Asana-Math, Asana;background:#FFF;" width="660" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:13.5px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="126.95999145507812" y="146.87998962402344">triangle circumscribed</text></g></g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:13.5px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="152.45498657226562" y="165.7799835205078">about a circle</text></g></g></g><g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:13.5px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="347.9599914550781" y="146.0550079345703">hexagon circumscribed</text></g></g><g><g><text style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:13.5px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);" x="377.2149963378906" y="164.9550018310547">about a circle</text></g></g></g></svg> </svg></center>
<p><strong><span style="color:#e74c3c;">NOTE</span></strong>: You DON'T really need a circle. For example, you can inscribe a square inside another square, or a square inside a hexagon, or something like that (see below).</p>
<center><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0" y="0" width="661.6000366210938" height="215.21250915527344" style="
width:661.6000366210938px;
height:215.21250915527344px;
background: #FFF;
fill: none;
">
<svg xmlns="http://www.w3.org/2000/svg" class="role-diagram-draw-area"><g class="shapes-region" style="stroke: black; fill: none;"><g class="composite-shape"><path class="real" d=" M259,9.24 L408.76,9.24 L408.76,159 L259,159 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"/></g><g class="composite-shape"><path class="real" d=" M327.09,9.95 L408.05,77.33 L340.67,158.29 L259.71,90.91 Z" style="stroke-width: 1; stroke: rgb(74, 144, 226); fill: none; fill-opacity: 1; stroke-opacity: 1;"/></g><g/></g><g/><g/><g/></svg>
<svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="660" height="213.6125030517578" style="width:660px;height:213.6125030517578px;font-family:Asana-Math, Asana;background:#FFF;"><g><g><g><text x="271.8000183105469" y="166.79998779296875" style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:15px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);">square inscribed in</text></g></g><g><g><text x="284.3000183105469" y="187.79998779296875" style="white-space:pre;stroke:none;fill:rgb(0,0,0);fill-opacity:1;font-size:15px;font-family:Arial, Helvetica, sans-serif;font-weight:400;font-style:normal;dominant-baseline:text-before-edge;text-decoration:none solid rgb(0, 0, 0);">another square</text></g></g></g></svg>
</svg>
</center>