Perimeter of a Square Inside a Circle
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In the figure shown, point $O$ is the center of the circle. One vertex of the square lies on the circle, and the opposite vertex is point $O$. If the area of the shaded region is $36\pi - 18$, what is the perimeter of the square?
A) 24
B) 72
C) $12\sqrt{2}$
D) $36\sqrt{2}$
This question includes visual content: A diagram shows a circle with center point $O$. A square is placed such that its bottom-right vertex is at the center $O$ and its top-left vertex lies on the circumference of the circle. The area within the circle that is outside the square is shaded gray. The square itself is white (unshaded). There is a small dot at the vertex on the circumference and at center $O$.
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Step by Step Written Solution
Let's find the perimeter of the square shown in this geometry problem. We are given a shaded region within a circle and a square centered at O.
Problem Analysis
Observe the figure. The shaded area is the area of the entire circle minus the area of the square. We are told this shaded area equals thirty-six pi minus eighteen.
Let the side length of the square be s. Then the area of the square is s squared.
Now let's look at the relationship between the square and the circle. The center of the circle is point O, and the opposite vertex of the square lies on the circumference. This means the diagonal of the square is equal to the radius of the circle.
The diagonal of a square with side s is s times the square root of 2. So, the radius r equals s square root of 2.
Now we can express the area of the circle in terms of s. Since the area is pi times r squared, we substitute s root two for r.
Setting up the Equation
When we square r, s squared multiplied by root two squared is just two s squared. So the circle area is two pi s squared.
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