Concentric Circles

<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-concentric-circles" target="_blank">Concentric Circles Exercise</a></p><p>In terms of math concepts, they don&#39;t come much simpler than <strong><span style="color:#8e44ad;">Concentric Circles</span></strong>.</p> <p style="margin-left: 40px;"><strong>Definition</strong>: Two (or more) circles who share the same center. They&#39;re very nice circles.</p> <p><strong><span style="color:#27ae60;">For You Visual Learners</span></strong></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="166.6999969482422" style=" width:661.6000366210938px; height:166.6999969482422px; 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=" M260,82.7 C260,41.67 293.27,8.4 334.3,8.4 C375.33,8.4 408.6,41.67 408.6,82.7 C408.6,123.73 375.33,157 334.3,157 C293.27,157 260,123.73 260,82.7 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"/></g><g class="composite-shape"><path class="real" d=" M286.5,82.7 C286.5,56.3 307.9,34.9 334.3,34.9 C360.7,34.9 382.1,56.3 382.1,82.7 C382.1,109.1 360.7,130.5 334.3,130.5 C307.9,130.5 286.5,109.1 286.5,82.7 Z" style="stroke-width: 1; 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fill: none;"><g class="composite-shape"><path class="real" d=" M260,82.7 C260,41.67 293.27,8.4 334.3,8.4 C375.33,8.4 408.6,41.67 408.6,82.7 C408.6,123.73 375.33,157 334.3,157 C293.27,157 260,123.73 260,82.7 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=" M318.65,82.7 C318.65,74.06 325.66,67.05 334.3,67.05 C342.94,67.05 349.95,74.06 349.95,82.7 C349.95,91.34 342.94,98.35 334.3,98.35 C325.66,98.35 318.65,91.34 318.65,82.7 Z" style="stroke-width: 1; stroke: rgb(75, 0, 130); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M332,82.7 C332,81.43 333.03,80.4 334.3,80.4 C335.57,80.4 336.6,81.43 336.6,82.7 C336.6,83.97 335.57,85 334.3,85 C333.03,85 332,83.97 332,82.7 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: rgb(0, 0, 0); fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M310.22,82.7 C310.22,69.4 321,58.62 334.3,58.62 C347.6,58.62 358.38,69.4 358.38,82.7 C358.38,96 347.6,106.78 334.3,106.78 C321,106.78 310.22,96 310.22,82.7 Z" style="stroke-width: 1; stroke: rgb(143, 0, 255); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M295.51,82.7 C295.51,61.28 312.88,43.91 334.3,43.91 C355.72,43.91 373.09,61.28 373.09,82.7 C373.09,104.12 355.72,121.49 334.3,121.49 C312.88,121.49 295.51,104.12 295.51,82.7 Z" style="stroke-width: 1; stroke: rgb(126, 211, 33); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M283.66,82.7 C283.66,54.73 306.33,32.06 334.3,32.06 C362.27,32.06 384.94,54.73 384.94,82.7 C384.94,110.67 362.27,133.34 334.3,133.34 C306.33,133.34 283.66,110.67 283.66,82.7 Z" style="stroke-width: 1; stroke: rgb(248, 231, 28); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M303.51,82.7 C303.51,65.7 317.3,51.91 334.3,51.91 C351.3,51.91 365.09,65.7 365.09,82.7 C365.09,99.7 351.3,113.49 334.3,113.49 C317.3,113.49 303.51,99.7 303.51,82.7 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=" M272.76,82.7 C272.76,48.71 300.31,21.16 334.3,21.16 C368.29,21.16 395.84,48.71 395.84,82.7 C395.84,116.69 368.29,144.24 334.3,144.24 C300.31,144.24 272.76,116.69 272.76,82.7 Z" style="stroke-width: 1; stroke: rgb(245, 166, 35); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g></g></g><g></g><g></g><g></g></svg> <svg height="165.10000610351562" style="width:660px;height:165.10000610351562px;font-family:Asana-Math, Asana;background:#FFF;" width="660" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"></svg> </svg></center> <p style="text-align: center;"><em>I dare you to stare at that for a minute straight.</em></p> <p><strong><span style="color:#e74c3c;">Common Problem with Concentric Circles</span></strong></p> <p>What is the area of the shaded region below?</p> <center><svg height="166.6999969482422" style=" width:661.6000366210938px; height:166.6999969482422px; 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=" M260,82.7 C260,41.67 293.27,8.4 334.3,8.4 C375.33,8.4 408.6,41.67 408.6,82.7 C408.6,123.73 375.33,157 334.3,157 C293.27,157 260,123.73 260,82.7 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: rgb(192, 192, 192); fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M332,82.7 C332,81.43 333.03,80.4 334.3,80.4 C335.57,80.4 336.6,81.43 336.6,82.7 C336.6,83.97 335.57,85 334.3,85 C333.03,85 332,83.97 332,82.7 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: rgb(0, 0, 0); fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M283.66,82.7 C283.66,54.73 306.33,32.06 334.3,32.06 C362.27,32.06 384.94,54.73 384.94,82.7 C384.94,110.67 362.27,133.34 334.3,133.34 C306.33,133.34 283.66,110.67 283.66,82.7 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: rgb(255, 255, 255); fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M331.92,82.32 C331.92,81 332.98,79.93 334.3,79.93 C335.62,79.93 336.68,81 336.68,82.32 C336.68,83.63 335.62,84.7 334.3,84.7 C332.98,84.7 331.92,83.63 331.92,82.32 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: rgb(0, 0, 0); fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M334.3,82.32 L334.3,32.06" stroke-dasharray="1.125 3.35" style="stroke: rgb(208, 2, 27); stroke-width: 1; fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M260,82.7 L334.3,82.32" stroke-dasharray="1.125 3.35" style="stroke: rgb(208, 2, 27); stroke-width: 1; fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g></g></g><g></g><g></g><g></g></svg> <svg height="165.10000610351562" style="width:660px;height:165.10000610351562px;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><g transform="matrix(1,0,0,1,338.8000183105469,63.400001525878906)"><path d="M280 181L280 106C280 46 269 32 220 30L158 27L158 -3C291 0 291 0 315 0C339 0 339 0 472 -3L472 27L424 30C375 33 364 46 364 106L364 181C423 181 444 180 472 177L472 248L364 245L365 697L285 667L2 204L2 181ZM280 245L65 245L280 597Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,292.4562683105469,99.0999984741211)"><path d="M131 331C152 512 241 611 421 665L379 689C283 657 242 637 191 593C88 506 32 384 32 247C32 82 112 -20 241 -20C371 -20 468 83 468 219C468 334 399 409 293 409C216 409 184 370 131 331ZM255 349C331 349 382 283 382 184C382 80 331 13 254 13C169 13 123 86 123 220C123 255 127 274 138 291C160 325 207 349 255 349Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.017,0,0,-0.017,0,0)"></path></g></g></g></g></svg> </svg></center> <p>To solve, simply subtract the area of the smaller circle from the area of the larger:</p> <p>$$\pi R^2 - \pi r^2$$</p> <p>$$36\pi - 16\pi$$</p> <p><strong><span style="color:#27ae60;">$$20\pi$$</span></strong></p>