Inscribed Squares

A square is inscribed in a circle or a polygon if its four vertices lie on the circumference of the circle or on the sides of the polygon. Figure A shows a square inscribed in a circle. Figure B shows a square inscribed in a triangle. Figure C shows a square inscribed in a quadrilateral.

Square \(ABCD\) is inscribed in a circle with center at \(O,\) as shown in the figure. If the area of the shaded region is \(25\pi -50\), find the area of the square.

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SOLUTION

Let \(r\) be the radius of the circle, and \(x\) the side length of the square, then the area of the square is \(x^2\). By the Pythagorean theorem, we have \((2r)^2=x^2+x^2.\)

Simplifying further, we get \(x^2=2r^2.\qquad (1)\)

The area of the shaded region is equal to the area of the circle minus the area of the square, so we have

\[\begin{align} 25\pi -50 &=\pi r^2 - 2r^2\\ &=r^2(\pi-2)\\ r^2&=\dfrac{25\pi -50}{\pi -2}\\ &=25.\qquad (2) \end{align}\]

Now substituting (2) into (1) gives \(x^2=2\times 25=50.\ _\square \)

A square with side length \(a\) is inscribed in a circle. Express the radius of the circle in terms of \(a\).

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SOLUTION

The diagonal of the square is the diameter of the circle. Let \(d\) and \(r\) be the diameter and radius of the circle, respectively. Then by the Pythagorean theorem, we have

\[\begin{align} d^2&=a^2+a^2\\ &=2a^2\\ d&=\sqrt{2a^2}\\ &=a\sqrt{2}. \end{align}\]

We know that the diameter is twice the radius, so

\[r=\dfrac{d}{2}=\dfrac{a\sqrt{2}}{2}.\ _\square \]