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A square is circumscribed within a circle, which is in turn circumscribed within another square as shown below.
Which area is larger - area "A" or area "B"?
If a side of the largest square has length "S," then the largest square has an area of S2. Also, the circle has an area of Π/4S2.
The radius of the circle is 1/2 S. The length of a side of the smaller square is 2a. The length of a can be found using the Pythagorean theorem:
a2 + a2= (1/2 S)2.
Solving for a yields a = S  1/8. Therefore, the area of the small square is (2a)2= (2S 1/8)2= 1/2 S2.
The area "A" is equal to 1/4 (S2 - Π/4S2) = 1/4S2 (1- Π/4), and the area "B" is equal to 1/4(Π/4S2 - 1/2 S2) = 1/4 S2 (Π/4 -1/2). Therefore, the area "B" is larger than the area "A". |
