It's All the Same Formula

<p>Notice that&nbsp;<strong><span style="color:#8e44ad;">It&#39;s All the Same Formula&nbsp;</span></strong>when finding the area of a triangle or quadrilateral.</p> <p>$$(the \ average \ of \ the \ top \ and \ bottom) \times height$$</p> <p><strong><span style="color:#e74c3c;">Squares, Rectangles, Parallelograms, and Rhombuses</span></strong></p> <p>In the case of these polygons, the top and the bottom are the same, so the average is simply the bottom (or the top). And the formula is:</p> <p>$$base \times height$$</p> <p><strong><span style="color:#e74c3c;">Trapezoids</span></strong></p> <p>Of course trapezoids have the average of the top and bottom embedded into the formula itself:</p> <p>$$(average \ of \ the \ bases) \times height = \frac{base_1 + base_2}{2} \times height$$</p> <p><strong><span style="color:#e74c3c;">But It Even Works for Triangles!</span></strong></p> <center><svg height="201.08570861816406" style=" width:661.8285522460938px; height:201.08570861816406px; background: #FFF; fill: none; " width="661.8285522460938" 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=" M330.3,34.26 L433.6,163 L227,163 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M330.3,34.26 L330.6,162.26" stroke-dasharray="6 6" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M330.6,150.86 L342,150.86 L342,162.26" 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="199.25714111328125" style="width:660px;height:199.25714111328125px;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(1,0,0,1,325.91424560546875,26.857138061523436)"><path d="M263 689C108 689 29 566 29 324C29 207 50 106 85 57C120 8 176 -20 238 -20C389 -20 465 110 465 366C465 585 400 689 263 689ZM245 654C342 654 381 556 381 316C381 103 343 15 251 15C154 15 113 116 113 360C113 571 150 654 245 654Z" fill="rgb(208,2,27)" fill-opacity="1" stroke="rgb(208,2,27)" stroke-opacity="1" stroke-width="8" transform="matrix(0.020399999999999998,0,0,-0.020399999999999998,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,325.91424560546875,183.8571533203125)"><path d="M235 722L223 733C171 707 135 698 63 691L59 670L107 670C131 670 141 663 141 646C141 639 140 628 139 622L38 71C37 68 37 64 37 61C37 22 85 -11 140 -11C177 -11 228 8 271 39C367 107 433 244 433 376C433 414 424 453 412 468C405 477 392 482 377 482C353 482 323 474 295 460C244 433 211 403 149 324ZM322 424C348 424 361 401 361 352C361 288 340 202 310 137C277 68 237 36 183 36C137 36 112 59 112 101C112 135 127 276 208 361C241 395 293 424 322 424Z" fill="rgb(208,2,27)" fill-opacity="1" stroke="rgb(208,2,27)" stroke-opacity="1" stroke-width="8" transform="matrix(0.020399999999999998,0,0,-0.020399999999999998,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,337.91424560546875,109.28571166992188)"><path d="M236 722L224 733C179 711 138 697 64 691L60 670L108 670C126 670 142 667 142 647C142 641 142 632 140 622L98 388C78 272 36 80 10 2L17 -9L86 7C94 64 108 164 148 236C193 317 296 414 338 414C349 414 360 407 360 393C360 375 355 342 345 303L294 107C288 85 281 55 281 31C281 6 291 -9 312 -9C344 -9 412 41 471 85L461 103L435 86C412 71 386 56 374 56C367 56 361 65 361 76C361 88 364 101 368 116L432 372C438 398 443 423 443 447C443 464 437 482 411 482C376 482 299 437 231 374C198 343 172 308 144 273L140 275Z" fill="rgb(208,2,27)" fill-opacity="1" stroke="rgb(208,2,27)" stroke-opacity="1" stroke-width="8" transform="matrix(0.020399999999999998,0,0,-0.020399999999999998,0,0)"></path></g></g></g></g></svg> </svg></center> <p>Notice how the bottom is $b$ and the top has a &quot;length&quot; of $0$. So, if we find the average of the bottom and top ($b$ and $0$) and then multiply it by the height, do we get the more familiar triangle area formula? Let&#39;s see.</p> <p>$$\frac{b+0}{2} \times h = \frac{b}{2} \times h = \frac{bh}{2}$$</p> <p>MIND BLOWNNNNNNNNNNNNNNNNNNNNNNNNNNNNN</p>