Longest Diagonal of Cube
<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-longest-diagonal-of-cube" target="_blank">Longest Diagonal of Cube Exercise</a></p><p><strong><span style="color:#8e44ad;">The Longest Diagonal of a Cube</span></strong> is this guy.</p>
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<p><strong><span style="color:#e74c3c;">If You Want to Skip the Backstory</span></strong></p>
<p style="text-align: center;"><span style="font-size:22px;">longest diagonal of a cube $=\sqrt{3}x$</span></p>
<p><strong><span style="color:#27ae60;">I Want the Backstory. How Do We Find This Guy?</span></strong></p>
<p>Imagine we have a right triangle, outlined in red.</p>
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<p>We know the height of this right triangle is $x$. If we find the base, we can then do Pythagorean theorem to find the hypotenuse, which just so happens to be the longest diagonal of the cube.</p>
<p><strong><span style="color:#27ae60;">But How Do We Find the Base?</span></strong></p>
<p>Imagine another right triangle, this time on the bottom of the cube. Notice how the base is the hypotenuse of a $45$-$45$-$90$ triangle, which is equal to $\sqrt{2}x$.</p>
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<p><strong><span style="color:#27ae60;">Put It All Together</span></strong></p>
<p>Now that we have our base, we can solve another Pythagorean theorem problem to get the longest diagonal of the cube.</p>
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<p>$$(\sqrt{2}x)^2 + x^2 = hypotenuse^2$$</p>
<p>$$2x^2 + x^2 = hypotenuse^2$$</p>
<p>$$3x^2 = hypotenuse^2$$</p>
<p>$$\sqrt{3}x = hypotenuse$$</p>