Max Area of Inscribed Polygon
<p><a href="https://www.prepswift.com/quizzes/quiz/prepswift-max-area-of-inscribed-polygon" target="_blank">Max Area of Inscribed Polygon Exercise</a></p><p>Do you remember our mountain entry on maximizing the area of a polygon? Recall that we would make the shape regular.</p>
<p>The same applies here. The <strong><span style="color:#8e44ad;">Max Area of an Inscribed Polygon</span></strong> is its regular shape. </p>
<p><strong><span style="color:#27ae60;">Example</span></strong></p>
<p>Imagine points $A$ and $C$ in the diagram below are maneuverable around the circumference of the circle. What shape produces the largest area of the inscribed quadrilateral?</p>
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<p>It's a Square! If we move points $A$ and $C$ so that inscribed quadrilateral $ABCD$ is a square, we have maximized its area. The same would apply for a triangle, pentagon, hexagon, septagon, etc.</p>
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<p><strong><span style="color:#e74c3c;">Additional Problem</span></strong></p>
<p>What is the area of the largest inscribed quadrilateral in a circle with a radius of $6$?</p>
<p>We already know it's a square, so let's see what it looks like.</p>
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<p><strong><span style="color:#e74c3c;">The Solution</span></strong></p>
<ul>
<li>In <strong><span style="color:#4a90e2;">Step 1</span></strong>, we simply visualize what the question is asking us. We inscribe a square inside a circle and label the radius as $6$. </li>
<li>In <strong><span style="color:#4a90e2;">Step 2</span></strong>, we realize that, if $6$ is the radius, then $12$ is the diameter, which is ALSO the diagonal of the square.</li>
<li>In <strong><span style="color:#4a90e2;">Step 3</span></strong>, we use our knowledge of $45$-$45$-$90$ triangles, and realize that the measure of each side of the inscribed square is $\frac{12}{\sqrt{2}}$.</li>
<li>Finally, we solve (see below):</li>
</ul>
<p>$$\frac{12}{\sqrt{2}} \times \frac{12}{\sqrt{2}} = \frac{144}{2} = 72$$</p>
<p>There's another thing to consider in this <strong><span style="color:#8e44ad;">Max Area of Inscribed Polygon II</span></strong> mountain entry.</p>
<p>What if a square is inscribed in another square? How do we maximize its area? See if you can figure out with this diagram below.</p>
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<p>Can you see what's happening? As we rotate the square to the right, the area has to get bigger for it to stay inscribed. It's not super obvious from the first to the second panel, but, as you can see, the third panel dramatically shows the area increase. </p>
<p>To maximize the area of an inscribed square in another square, you simply have to make the inscribed square look more and more like the original square. This means that the <strong><span style="color:#e74c3c;">SMALLEST </span></strong>possible area for a square inscribed in another square is when the inscribed square's vertices are at the midpoints of the sides of the larger square.</p>
<p>If that made no sense to you, don't worry. It was a hell of a sentence.</p>
<p>In a nutshell, the smallest you can inscribe a square in another square is when the inside square is exactly half the area of the larger square.</p>