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by alex.gellatly » Mon Apr 16, 2012 11:35 pm
A square is inscribed in a circle. If the area of the square region is 16, what is the area of the circular region?

(A) 2pi
(B) 4pi
(C) 6pi
(D) 12pi
(E) 16pi

Thanks
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by Anurag@Gurome » Mon Apr 16, 2012 11:44 pm
alex.gellatly wrote:A square is inscribed in a circle. If the area of the square region is 16, what is the area of the circular region?

(A) 2pi
(B) 4pi
(C) 6pi
(D) 12pi
(E) 16pi

Thanks

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Area of the square region is 16.
If each side of the square = s, then area of square = s²
s² 16 implies s = 4
Now we can find length of diagonal AC, which is the diameter of the circle.
By Pythagoras Theorem, AC² = AB² + BC²
AC² = 4² + 4² = 32
AC = √32 = 4√2

This means radius of circle = 2√2
Area of circle = (pi) * radius² = (pi) * (2√2)² = [spoiler]8(pi) [/spoiler]sq units, which is not given in the answer choices.
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by aneesh.kg » Mon Apr 16, 2012 11:46 pm
If the area of the square is 16, then each side of the square must be 4.

Please note that, the diameter of the circle which is also the diagonal of a square.

Using Pythagoras' Theorem to find the length of the diagonal of the square,

d^2 = 4^2 + 4^2
or
d = 4 . (2)^0.5
or
r = 2 . (2)^0.5

Area of the circle = pi.(r)^2
= pi . 8

The answer is 8pi (which is surprisingly not in the options)
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