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In the figure above, a square is inscribed in a circle with diameter d. What is the sum of the areas of the shaded regions, in terms of d ?

• (A)    CORRECT ANSWER
• (B)
• (C)
• (D)
• (E)

Explanation:

The sum of the areas of the shaded regions is the area of the circle minus the area of the square. The area of the circle is where Therefore, this area is . To find the area of the square, you need the length of one of the sides. Since the diagonal of the square is d, each side equals . Therefore, the area of the square is . The sum of the areas of the shaded regions is which equals . The correct answer is (A).

(from the October 12, 1999 test)