Longest Diagonal of Rectangular Solid
<p><span style="font-size:22px"><a href="https://www.prepswift.com/quizzes/quiz/prepswift-longest-diagonal-of-rectangular-solid" target="_blank">Longest Diagonal of Rectangular Solid Exercise</a></span></p><p>The <strong><span style="color:#8e44ad;">Longest Diagonal of a Rectangular Solid</span></strong> is this guy.</p>
<center><svg height="201.67857360839844" style="
width:661.8285522460938px;
height:201.67857360839844px;
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=" M192,58.24 L233.04,17.2 L476.6,17.2 L476.6,112.96 L435.56,154 L192,154 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path><path class="real" d=" M476.6,17.2 L435.56,58.24 L192,58.24" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path><path class="real" d=" M435.56,58.24 L435.56,154" 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=" M233.04,17.2 L435.56,154" stroke-dasharray="" 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=" M476.6,112.96 L233.04,111.56" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M233.04,17.2 L233.04,111.56" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M233.04,111.56 L192,154" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g></g></g><g></g><g></g><g></g></svg> <svg height="199.85000610351562" style="width:660px;height:199.85000610351562px;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(1,0,0,1,306.91424560546875,175.9371304321289)"><path d="M9 1C24 -7 40 -11 52 -11C85 -11 124 18 155 65L231 182L242 113C255 28 278 -11 314 -11C336 -11 368 6 400 35L449 79L440 98C404 68 379 53 363 53C348 53 335 63 325 83C316 102 305 139 300 168L282 269L317 318C364 383 391 406 422 406C438 406 450 398 455 383L469 387L484 472C472 479 463 482 454 482C414 482 374 446 312 354L275 299L269 347C257 446 230 482 171 482C145 482 123 474 114 461L56 378L73 368C103 402 123 416 142 416C175 416 197 375 214 277L225 215L185 153C142 86 108 54 80 54C65 54 54 58 52 63L41 91L21 88C21 53 13 27 9 1Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,461.914306640625,144.8657192993164)"><path d="M-7 -180C-8 -187 -8 -193 -8 -198C-8 -241 29 -276 74 -276C180 -276 290 -152 349 33L490 473L479 482C450 471 427 465 405 463L370 331C358 284 323 211 290 162C255 111 206 67 184 67C172 67 163 90 164 115L180 322C182 353 184 391 184 419C184 464 177 482 160 482C147 482 133 475 85 442L3 386L14 368L64 398C69 401 80 410 89 410C103 410 111 391 111 358C111 357 111 351 110 343L93 100L92 60C92 18 110 -11 135 -11C172 -11 256 74 331 187L282 16C231 -161 181 -234 111 -234C76 -234 49 -207 49 -172C49 -167 50 -159 51 -150L41 -146Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g></g></g><g><g><g><g transform="matrix(1,0,0,1,480.914306640625,69.86571929931641)"><path d="M322 -11C334 -11 343 -9 372 -1C383 61 391 95 402 142L380 142L358 82C352 64 340 57 319 57C307 57 291 59 270 64L252 68C219 75 185 80 166 80C134 80 107 74 71 59L410 441L416 473L407 482L384 459C374 449 364 445 350 445C316 445 264 453 216 466L198 471C170 478 150 482 136 482C119 482 99 479 75 472L43 350L64 350L94 408C111 412 120 413 133 413C174 413 206 398 259 398C295 398 319 404 352 421L-1 21L8 -6C37 17 64 26 99 26C162 26 253 -11 322 -11Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.024480000000000002,0,0,-0.024480000000000002,0,0)"></path></g></g></g></g></svg> </svg></center>
<p>To find the distance of this diagonal, much like in the previous mountain entry, picture two right triangles, one on the bottom side and one running through the middle of the solid.</p>
<center><svg height="153.51499938964844" style="
width:661.9199829101562px;
height:153.51499938964844px;
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=" M366.6,44.91 L398.73,12.78 L589.38,12.78 L589.38,87.74 L557.25,119.86 L366.6,119.86 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path><path class="real" d=" M589.38,12.78 L557.25,44.91 L366.6,44.91" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path><path class="real" d=" M557.25,44.91 L557.25,119.86" 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=" M398.73,12.78 L557.25,119.86" stroke-dasharray="" 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=" M589.38,87.74 L398.73,86.64" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M398.73,12.78 L398.73,86.64" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M398.73,86.64 L366.6,119.86" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M398.73,12.78 L398.73,86.64" stroke-dasharray="" style="stroke: rgb(208, 2, 27); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M557.25,119.86 L398.73,86.64" stroke-dasharray="" style="stroke: rgb(208, 2, 27); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M48.6,44.91 L80.73,12.78 L271.38,12.78 L271.38,87.74 L239.25,119.86 L48.6,119.86 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path><path class="real" d=" M271.38,12.78 L239.25,44.91 L48.6,44.91" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path><path class="real" d=" M239.25,44.91 L239.25,119.86" 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=" M271.38,87.74 L80.73,86.64" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M80.73,12.78 L80.73,86.64" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M80.73,86.64 L48.6,119.86" stroke-dasharray="" 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=" M80.73,12.78 L80.73,86.64" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M239.25,119.86 L80.73,86.64" stroke-dasharray="" style="stroke: rgb(208, 2, 27); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M48.6,119.86 L239.25,119.86" stroke-dasharray="" 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="151.59500122070312" style="width:660px;height:151.59500122070312px;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(1,0,0,1,69.58499145507812,57.16000213623047)"><path d="M322 -11C334 -11 343 -9 372 -1C383 61 391 95 402 142L380 142L358 82C352 64 340 57 319 57C307 57 291 59 270 64L252 68C219 75 185 80 166 80C134 80 107 74 71 59L410 441L416 473L407 482L384 459C374 449 364 445 350 445C316 445 264 453 216 466L198 471C170 478 150 482 136 482C119 482 99 479 75 472L43 350L64 350L94 408C111 412 120 413 133 413C174 413 206 398 259 398C295 398 319 404 352 421L-1 21L8 -6C37 17 64 26 99 26C162 26 253 -11 322 -11Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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(1,0,0,1,53.584991455078125,99.16000213623047)"><path d="M-7 -180C-8 -187 -8 -193 -8 -198C-8 -241 29 -276 74 -276C180 -276 290 -152 349 33L490 473L479 482C450 471 427 465 405 463L370 331C358 284 323 211 290 162C255 111 206 67 184 67C172 67 163 90 164 115L180 322C182 353 184 391 184 419C184 464 177 482 160 482C147 482 133 475 85 442L3 386L14 368L64 398C69 401 80 410 89 410C103 410 111 391 111 358C111 357 111 351 110 343L93 100L92 60C92 18 110 -11 135 -11C172 -11 256 74 331 187L282 16C231 -161 181 -234 111 -234C76 -234 49 -207 49 -172C49 -167 50 -159 51 -150L41 -146Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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(1,0,0,1,136.58499145507812,136.16000213623047)"><path d="M9 1C24 -7 40 -11 52 -11C85 -11 124 18 155 65L231 182L242 113C255 28 278 -11 314 -11C336 -11 368 6 400 35L449 79L440 98C404 68 379 53 363 53C348 53 335 63 325 83C316 102 305 139 300 168L282 269L317 318C364 383 391 406 422 406C438 406 450 398 455 383L469 387L484 472C472 479 463 482 454 482C414 482 374 446 312 354L275 299L269 347C257 446 230 482 171 482C145 482 123 474 114 461L56 378L73 368C103 402 123 416 142 416C175 416 197 375 214 277L225 215L185 153C142 86 108 54 80 54C65 54 54 58 52 63L41 91L21 88C21 53 13 27 9 1Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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(1,0,0,1,452.5849914550781,135.16000213623047)"><path d="M9 1C24 -7 40 -11 52 -11C85 -11 124 18 155 65L231 182L242 113C255 28 278 -11 314 -11C336 -11 368 6 400 35L449 79L440 98C404 68 379 53 363 53C348 53 335 63 325 83C316 102 305 139 300 168L282 269L317 318C364 383 391 406 422 406C438 406 450 398 455 383L469 387L484 472C472 479 463 482 454 482C414 482 374 446 312 354L275 299L269 347C257 446 230 482 171 482C145 482 123 474 114 461L56 378L73 368C103 402 123 416 142 416C175 416 197 375 214 277L225 215L185 153C142 86 108 54 80 54C65 54 54 58 52 63L41 91L21 88C21 53 13 27 9 1Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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(1,0,0,1,386.5849914550781,60.16000213623047)"><path d="M322 -11C334 -11 343 -9 372 -1C383 61 391 95 402 142L380 142L358 82C352 64 340 57 319 57C307 57 291 59 270 64L252 68C219 75 185 80 166 80C134 80 107 74 71 59L410 441L416 473L407 482L384 459C374 449 364 445 350 445C316 445 264 453 216 466L198 471C170 478 150 482 136 482C119 482 99 479 75 472L43 350L64 350L94 408C111 412 120 413 133 413C174 413 206 398 259 398C295 398 319 404 352 421L-1 21L8 -6C37 17 64 26 99 26C162 26 253 -11 322 -11Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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(1,0,0,1,372.5849914550781,98.16000213623047)"><path d="M-7 -180C-8 -187 -8 -193 -8 -198C-8 -241 29 -276 74 -276C180 -276 290 -152 349 33L490 473L479 482C450 471 427 465 405 463L370 331C358 284 323 211 290 162C255 111 206 67 184 67C172 67 163 90 164 115L180 322C182 353 184 391 184 419C184 464 177 482 160 482C147 482 133 475 85 442L3 386L14 368L64 398C69 401 80 410 89 410C103 410 111 391 111 358C111 357 111 351 110 343L93 100L92 60C92 18 110 -11 135 -11C172 -11 256 74 331 187L282 16C231 -161 181 -234 111 -234C76 -234 49 -207 49 -172C49 -167 50 -159 51 -150L41 -146Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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>In the bottom right triangle, we can do the Pythagorean theorem to find the hypotenuse, which is conveniently the base of the right triangle running through the middle of the solid. We can then do Pythagorean Theorem again to find the longest diagonal in a rectangular solid.</p>
<center><svg height="153.51499938964844" style="
width:661.9199829101562px;
height:153.51499938964844px;
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=" M366.6,44.91 L398.73,12.78 L589.38,12.78 L589.38,87.74 L557.25,119.86 L366.6,119.86 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path><path class="real" d=" M589.38,12.78 L557.25,44.91 L366.6,44.91" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path><path class="real" d=" M557.25,44.91 L557.25,119.86" 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=" M398.73,12.78 L557.25,119.86" stroke-dasharray="" 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=" M589.38,87.74 L398.73,86.64" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M398.73,12.78 L398.73,86.64" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M398.73,86.64 L366.6,119.86" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M398.73,12.78 L398.73,86.64" stroke-dasharray="" style="stroke: rgb(208, 2, 27); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M557.25,119.86 L398.73,86.64" stroke-dasharray="" style="stroke: rgb(208, 2, 27); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M48.6,44.91 L80.73,12.78 L271.38,12.78 L271.38,87.74 L239.25,119.86 L48.6,119.86 Z" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path><path class="real" d=" M271.38,12.78 L239.25,44.91 L48.6,44.91" style="stroke-width: 1; stroke: rgb(0, 0, 0); fill: none; fill-opacity: 1;"></path><path class="real" d=" M239.25,44.91 L239.25,119.86" 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=" M271.38,87.74 L80.73,86.64" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M80.73,12.78 L80.73,86.64" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M80.73,86.64 L48.6,119.86" stroke-dasharray="" 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=" M80.73,12.78 L80.73,86.64" stroke-dasharray="1.125 3.35" style="stroke: rgb(0, 0, 0); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M239.25,119.86 L80.73,86.64" stroke-dasharray="" style="stroke: rgb(208, 2, 27); stroke-opacity: 1; stroke-width: 1; fill: none; fill-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M48.6,119.86 L239.25,119.86" stroke-dasharray="" style="stroke: rgb(208, 2, 27); stroke-width: 1; fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="composite-shape"><path class="real" d=" M301,72.3 L321.42,72.3 L321.42,66.56 L335.04,78.04 L321.42,89.51 L321.42,83.78 L301,83.78 Z" style="stroke-width: 1; stroke: rgb(74, 144, 226); fill: none; fill-opacity: 1; stroke-opacity: 1;"></path></g><g class="arrow-line"><path class="connection real" d=" M492.04,39.56 L479.23,56.95" stroke-dasharray="" style="stroke: rgb(208, 2, 27); stroke-width: 1; fill: none; fill-opacity: 1; stroke-opacity: 1;"></path><g stroke="rgb(208,2,27)" stroke-opacity="1" style="stroke: rgb(208, 2, 27); stroke-width: 1;" transform="matrix(0.5931378895184068,-0.8051008905830686,0.8051008905830686,0.5931378895184068,478.0400085449219,58.56000518798828)"><path d=" M10.93,-3.29 Q4.96,-0.45 0,0 Q4.96,0.45 10.93,3.29"></path></g></g><g></g></g><g></g><g></g><g></g></svg> <svg height="151.59500122070312" style="width:660px;height:151.59500122070312px;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(1,0,0,1,69.58499145507812,57.159994506835936)"><path d="M322 -11C334 -11 343 -9 372 -1C383 61 391 95 402 142L380 142L358 82C352 64 340 57 319 57C307 57 291 59 270 64L252 68C219 75 185 80 166 80C134 80 107 74 71 59L410 441L416 473L407 482L384 459C374 449 364 445 350 445C316 445 264 453 216 466L198 471C170 478 150 482 136 482C119 482 99 479 75 472L43 350L64 350L94 408C111 412 120 413 133 413C174 413 206 398 259 398C295 398 319 404 352 421L-1 21L8 -6C37 17 64 26 99 26C162 26 253 -11 322 -11Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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(1,0,0,1,53.584991455078125,99.15999450683594)"><path d="M-7 -180C-8 -187 -8 -193 -8 -198C-8 -241 29 -276 74 -276C180 -276 290 -152 349 33L490 473L479 482C450 471 427 465 405 463L370 331C358 284 323 211 290 162C255 111 206 67 184 67C172 67 163 90 164 115L180 322C182 353 184 391 184 419C184 464 177 482 160 482C147 482 133 475 85 442L3 386L14 368L64 398C69 401 80 410 89 410C103 410 111 391 111 358C111 357 111 351 110 343L93 100L92 60C92 18 110 -11 135 -11C172 -11 256 74 331 187L282 16C231 -161 181 -234 111 -234C76 -234 49 -207 49 -172C49 -167 50 -159 51 -150L41 -146Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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(1,0,0,1,136.58499145507812,136.15997924804688)"><path d="M9 1C24 -7 40 -11 52 -11C85 -11 124 18 155 65L231 182L242 113C255 28 278 -11 314 -11C336 -11 368 6 400 35L449 79L440 98C404 68 379 53 363 53C348 53 335 63 325 83C316 102 305 139 300 168L282 269L317 318C364 383 391 406 422 406C438 406 450 398 455 383L469 387L484 472C472 479 463 482 454 482C414 482 374 446 312 354L275 299L269 347C257 446 230 482 171 482C145 482 123 474 114 461L56 378L73 368C103 402 123 416 142 416C175 416 197 375 214 277L225 215L185 153C142 86 108 54 80 54C65 54 54 58 52 63L41 91L21 88C21 53 13 27 9 1Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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(1,0,0,1,452.5849914550781,135.15997924804688)"><path d="M9 1C24 -7 40 -11 52 -11C85 -11 124 18 155 65L231 182L242 113C255 28 278 -11 314 -11C336 -11 368 6 400 35L449 79L440 98C404 68 379 53 363 53C348 53 335 63 325 83C316 102 305 139 300 168L282 269L317 318C364 383 391 406 422 406C438 406 450 398 455 383L469 387L484 472C472 479 463 482 454 482C414 482 374 446 312 354L275 299L269 347C257 446 230 482 171 482C145 482 123 474 114 461L56 378L73 368C103 402 123 416 142 416C175 416 197 375 214 277L225 215L185 153C142 86 108 54 80 54C65 54 54 58 52 63L41 91L21 88C21 53 13 27 9 1Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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(1,0,0,1,386.5849914550781,60.159994506835936)"><path d="M322 -11C334 -11 343 -9 372 -1C383 61 391 95 402 142L380 142L358 82C352 64 340 57 319 57C307 57 291 59 270 64L252 68C219 75 185 80 166 80C134 80 107 74 71 59L410 441L416 473L407 482L384 459C374 449 364 445 350 445C316 445 264 453 216 466L198 471C170 478 150 482 136 482C119 482 99 479 75 472L43 350L64 350L94 408C111 412 120 413 133 413C174 413 206 398 259 398C295 398 319 404 352 421L-1 21L8 -6C37 17 64 26 99 26C162 26 253 -11 322 -11Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" 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(1,0,0,1,372.5849914550781,98.15999450683594)"><path d="M-7 -180C-8 -187 -8 -193 -8 -198C-8 -241 29 -276 74 -276C180 -276 290 -152 349 33L490 473L479 482C450 471 427 465 405 463L370 331C358 284 323 211 290 162C255 111 206 67 184 67C172 67 163 90 164 115L180 322C182 353 184 391 184 419C184 464 177 482 160 482C147 482 133 475 85 442L3 386L14 368L64 398C69 401 80 410 89 410C103 410 111 391 111 358C111 357 111 351 110 343L93 100L92 60C92 18 110 -11 135 -11C172 -11 256 74 331 187L282 16C231 -161 181 -234 111 -234C76 -234 49 -207 49 -172C49 -167 50 -159 51 -150L41 -146Z" fill="rgb(0,0,0)" fill-opacity="1" stroke="rgb(0,0,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.020399999999999998,0,0,-0.020399999999999998,0,0)"></path></g></g></g></g><g transform="matrix(0.9824503977255098,0.1865240360087346,-0.1865240360087346,0.9824503977255098,19.789377581995268,-29.739274239649376)"><g><g><g><g><g><g transform="matrix(1,0,0,1,154.32748413085938,98.04999694824218)"><path d="M9 1C24 -7 40 -11 52 -11C85 -11 124 18 155 65L231 182L242 113C255 28 278 -11 314 -11C336 -11 368 6 400 35L449 79L440 98C404 68 379 53 363 53C348 53 335 63 325 83C316 102 305 139 300 168L282 269L317 318C364 383 391 406 422 406C438 406 450 398 455 383L469 387L484 472C472 479 463 482 454 482C414 482 374 446 312 354L275 299L269 347C257 446 230 482 171 482C145 482 123 474 114 461L56 378L73 368C103 402 123 416 142 416C175 416 197 375 214 277L225 215L185 153C142 86 108 54 80 54C65 54 54 58 52 63L41 91L21 88C21 53 13 27 9 1Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,161.90249633789062,93.0149951171875)"><path d="M16 23L16 -3C203 -3 203 0 239 0C275 0 275 -3 468 -3L468 82C353 77 307 81 122 77L304 270C401 373 431 428 431 503C431 618 353 689 226 689C154 689 105 669 56 619L39 483L68 483L81 529C97 587 133 612 200 612C286 612 341 558 341 473C341 398 299 324 186 204Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.009519999999999999,0,0,-0.009519999999999999,0,0)"></path></g></g></g></g></g><g><g transform="matrix(1,0,0,1,170.19247436523438,98.04999694824218)"><path d="M604 241L604 300L364 300L364 538L305 538L305 300L65 300L65 241L305 241L305 0L364 0L364 241Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,181.99746704101562,98.04999694824218)"><path d="M-7 -180C-8 -187 -8 -193 -8 -198C-8 -241 29 -276 74 -276C180 -276 290 -152 349 33L490 473L479 482C450 471 427 465 405 463L370 331C358 284 323 211 290 162C255 111 206 67 184 67C172 67 163 90 164 115L180 322C182 353 184 391 184 419C184 464 177 482 160 482C147 482 133 475 85 442L3 386L14 368L64 398C69 401 80 410 89 410C103 410 111 391 111 358C111 357 111 351 110 343L93 100L92 60C92 18 110 -11 135 -11C172 -11 256 74 331 187L282 16C231 -161 181 -234 111 -234C76 -234 49 -207 49 -172C49 -167 50 -159 51 -150L41 -146Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,189.57247924804688,93.0149951171875)"><path d="M16 23L16 -3C203 -3 203 0 239 0C275 0 275 -3 468 -3L468 82C353 77 307 81 122 77L304 270C401 373 431 428 431 503C431 618 353 689 226 689C154 689 105 669 56 619L39 483L68 483L81 529C97 587 133 612 200 612C286 612 341 558 341 473C341 398 299 324 186 204Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.009519999999999999,0,0,-0.009519999999999999,0,0)"></path></g></g></g></g></g></g></g><svg height="13.59999942779541" overflow="visible" width="54.41499710083008" x="140.72750854492188" y="82.4949951171875"><polygon points="0.28,8.91 3.83,7.01 7.70,15.22 13.09,0.00 54.41,0.00 54.41,0.84 13.69,0.84 7.34,18.75 2.48,8.55 0.51,9.29" style="fill:rgb(74,144,226);fill-opacity:1;stroke-width:1px;stroke:none;stroke-opacity:1;"></polygon></svg></g></g></g><g transform="matrix(0.9753037866844769,0.22086766100749183,-0.22086766100749183,0.9753037866844769,33.70218648111933,-92.83152577629863)"><g><g><g><g><g><g transform="matrix(1,0,0,1,418.3575134277344,112.04497680664062)"><path d="M9 1C24 -7 40 -11 52 -11C85 -11 124 18 155 65L231 182L242 113C255 28 278 -11 314 -11C336 -11 368 6 400 35L449 79L440 98C404 68 379 53 363 53C348 53 335 63 325 83C316 102 305 139 300 168L282 269L317 318C364 383 391 406 422 406C438 406 450 398 455 383L469 387L484 472C472 479 463 482 454 482C414 482 374 446 312 354L275 299L269 347C257 446 230 482 171 482C145 482 123 474 114 461L56 378L73 368C103 402 123 416 142 416C175 416 197 375 214 277L225 215L185 153C142 86 108 54 80 54C65 54 54 58 52 63L41 91L21 88C21 53 13 27 9 1Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,425.9324645996094,107.009990234375)"><path d="M16 23L16 -3C203 -3 203 0 239 0C275 0 275 -3 468 -3L468 82C353 77 307 81 122 77L304 270C401 373 431 428 431 503C431 618 353 689 226 689C154 689 105 669 56 619L39 483L68 483L81 529C97 587 133 612 200 612C286 612 341 558 341 473C341 398 299 324 186 204Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.009519999999999999,0,0,-0.009519999999999999,0,0)"></path></g></g></g></g></g><g><g transform="matrix(1,0,0,1,434.2225036621094,112.04497680664062)"><path d="M604 241L604 300L364 300L364 538L305 538L305 300L65 300L65 241L305 241L305 0L364 0L364 241Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,446.0274963378906,112.04497680664062)"><path d="M-7 -180C-8 -187 -8 -193 -8 -198C-8 -241 29 -276 74 -276C180 -276 290 -152 349 33L490 473L479 482C450 471 427 465 405 463L370 331C358 284 323 211 290 162C255 111 206 67 184 67C172 67 163 90 164 115L180 322C182 353 184 391 184 419C184 464 177 482 160 482C147 482 133 475 85 442L3 386L14 368L64 398C69 401 80 410 89 410C103 410 111 391 111 358C111 357 111 351 110 343L93 100L92 60C92 18 110 -11 135 -11C172 -11 256 74 331 187L282 16C231 -161 181 -234 111 -234C76 -234 49 -207 49 -172C49 -167 50 -159 51 -150L41 -146Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,453.6025085449219,107.009990234375)"><path d="M16 23L16 -3C203 -3 203 0 239 0C275 0 275 -3 468 -3L468 82C353 77 307 81 122 77L304 270C401 373 431 428 431 503C431 618 353 689 226 689C154 689 105 669 56 619L39 483L68 483L81 529C97 587 133 612 200 612C286 612 341 558 341 473C341 398 299 324 186 204Z" fill="rgb(74,144,226)" fill-opacity="1" stroke="rgb(74,144,226)" stroke-opacity="1" stroke-width="8" transform="matrix(0.009519999999999999,0,0,-0.009519999999999999,0,0)"></path></g></g></g></g></g></g></g><svg height="13.59999942779541" overflow="visible" width="54.41499710083008" x="404.7574768066406" y="96.489990234375"><polygon points="0.28,8.91 3.83,7.01 7.70,15.22 13.09,0.00 54.41,0.00 54.41,0.84 13.69,0.84 7.34,18.75 2.48,8.55 0.51,9.29" style="fill:rgb(74,144,226);fill-opacity:1;stroke-width:1px;stroke:none;stroke-opacity:1;"></polygon></svg></g></g></g><g><g><g><g><g><g><g transform="matrix(1,0,0,1,468.8550109863281,32.089990234375)"><path d="M9 1C24 -7 40 -11 52 -11C85 -11 124 18 155 65L231 182L242 113C255 28 278 -11 314 -11C336 -11 368 6 400 35L449 79L440 98C404 68 379 53 363 53C348 53 335 63 325 83C316 102 305 139 300 168L282 269L317 318C364 383 391 406 422 406C438 406 450 398 455 383L469 387L484 472C472 479 463 482 454 482C414 482 374 446 312 354L275 299L269 347C257 446 230 482 171 482C145 482 123 474 114 461L56 378L73 368C103 402 123 416 142 416C175 416 197 375 214 277L225 215L185 153C142 86 108 54 80 54C65 54 54 58 52 63L41 91L21 88C21 53 13 27 9 1Z" fill="rgb(88,163,0)" fill-opacity="1" stroke="rgb(88,163,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,476.4299621582031,27.054988403320312)"><path d="M16 23L16 -3C203 -3 203 0 239 0C275 0 275 -3 468 -3L468 82C353 77 307 81 122 77L304 270C401 373 431 428 431 503C431 618 353 689 226 689C154 689 105 669 56 619L39 483L68 483L81 529C97 587 133 612 200 612C286 612 341 558 341 473C341 398 299 324 186 204Z" fill="rgb(88,163,0)" fill-opacity="1" stroke="rgb(88,163,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.009519999999999999,0,0,-0.009519999999999999,0,0)"></path></g></g></g></g></g><g><g transform="matrix(1,0,0,1,484.7200012207031,32.089990234375)"><path d="M604 241L604 300L364 300L364 538L305 538L305 300L65 300L65 241L305 241L305 0L364 0L364 241Z" fill="rgb(88,163,0)" fill-opacity="1" stroke="rgb(88,163,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,496.5249938964844,32.089990234375)"><path d="M-7 -180C-8 -187 -8 -193 -8 -198C-8 -241 29 -276 74 -276C180 -276 290 -152 349 33L490 473L479 482C450 471 427 465 405 463L370 331C358 284 323 211 290 162C255 111 206 67 184 67C172 67 163 90 164 115L180 322C182 353 184 391 184 419C184 464 177 482 160 482C147 482 133 475 85 442L3 386L14 368L64 398C69 401 80 410 89 410C103 410 111 391 111 358C111 357 111 351 110 343L93 100L92 60C92 18 110 -11 135 -11C172 -11 256 74 331 187L282 16C231 -161 181 -234 111 -234C76 -234 49 -207 49 -172C49 -167 50 -159 51 -150L41 -146Z" fill="rgb(88,163,0)" fill-opacity="1" stroke="rgb(88,163,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,504.1000061035156,27.054988403320312)"><path d="M16 23L16 -3C203 -3 203 0 239 0C275 0 275 -3 468 -3L468 82C353 77 307 81 122 77L304 270C401 373 431 428 431 503C431 618 353 689 226 689C154 689 105 669 56 619L39 483L68 483L81 529C97 587 133 612 200 612C286 612 341 558 341 473C341 398 299 324 186 204Z" fill="rgb(88,163,0)" fill-opacity="1" stroke="rgb(88,163,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.009519999999999999,0,0,-0.009519999999999999,0,0)"></path></g></g></g></g></g><g><g transform="matrix(1,0,0,1,512.3899841308594,32.089990234375)"><path d="M604 241L604 300L364 300L364 538L305 538L305 300L65 300L65 241L305 241L305 0L364 0L364 241Z" fill="rgb(88,163,0)" fill-opacity="1" stroke="rgb(88,163,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g transform="matrix(1,0,0,1,524.1949768066406,32.089990234375)"><path d="M322 -11C334 -11 343 -9 372 -1C383 61 391 95 402 142L380 142L358 82C352 64 340 57 319 57C307 57 291 59 270 64L252 68C219 75 185 80 166 80C134 80 107 74 71 59L410 441L416 473L407 482L384 459C374 449 364 445 350 445C316 445 264 453 216 466L198 471C170 478 150 482 136 482C119 482 99 479 75 472L43 350L64 350L94 408C111 412 120 413 133 413C174 413 206 398 259 398C295 398 319 404 352 421L-1 21L8 -6C37 17 64 26 99 26C162 26 253 -11 322 -11Z" fill="rgb(88,163,0)" fill-opacity="1" stroke="rgb(88,163,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.0136,0,0,-0.0136,0,0)"></path></g></g><g><g><g><g><g transform="matrix(1,0,0,1,531.0049743652344,27.054988403320312)"><path d="M16 23L16 -3C203 -3 203 0 239 0C275 0 275 -3 468 -3L468 82C353 77 307 81 122 77L304 270C401 373 431 428 431 503C431 618 353 689 226 689C154 689 105 669 56 619L39 483L68 483L81 529C97 587 133 612 200 612C286 612 341 558 341 473C341 398 299 324 186 204Z" fill="rgb(88,163,0)" fill-opacity="1" stroke="rgb(88,163,0)" stroke-opacity="1" stroke-width="8" transform="matrix(0.009519999999999999,0,0,-0.009519999999999999,0,0)"></path></g></g></g></g></g></g></g><svg height="13.59999942779541" overflow="visible" width="81.31999969482422" x="455.2549743652344" y="16.534988403320312"><polygon points="0.28,8.91 3.83,7.01 7.70,15.22 13.09,0.00 81.32,0.00 81.32,0.84 13.69,0.84 7.34,18.75 2.48,8.55 0.51,9.29" style="fill:rgb(88,163,0);fill-opacity:1;stroke-width:1px;stroke:none;stroke-opacity:1;"></polygon></svg></g></g></g></svg> </svg></center>
<p><strong><span style="color:#27ae60;">The Formula</span></strong></p>
<p style="text-align: center;"><span style="font-size:22px;">longest diagonal <span style="color:#58a300;">$= \sqrt{x^2 + y^2 + z^2}$</span></span></p>