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.
Sample Problems
Square is inscribed in a circle with center at as shown in the figure. If the area of the shaded region is , find the area of the square.
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Let be the radius of the circle, and the side length of the square, then the area of the square is . By the Pythagorean theorem, we have
Simplifying further, we get
The area of the shaded region is equal to the area of the circle minus the area of the square, so we have
Now substituting (2) into (1) gives
A square with side length is inscribed in a circle. Express the radius of the circle in terms of .
The diagonal of the square is the diameter of the circle. Let and be the diameter and radius of the circle, respectively. Then by the Pythagorean theorem, we have
We know that the diameter is twice the radius, so