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4.3 Circles - Radius, diameter, circumference, area
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What is the length of one side of a square that has the same area as a circle with radius 2 ?

A.

2

B.

the square root of 2 pi, end root

C.

2 times the square root of pi

D.

2 pi

What is the length of one side of a square that has the same area as

Hard-difficulty · SAT Math · Circles — Radius, diameter, circumference, area. Read the question above, select your answer, and check the full explanation below to understand exactly why the correct choice works.

Answer explanation

Choice C is correct. The area A of a circle with radius r is given by the formula A equals, pi times r squared. Thus, a circle with radius 2 has area pi times 2 squared, which can be rewritten as 4 pi. The area of a square with side length s is given by the formula A equals s squared. Thus, if a square has the same area as a circle with radius 2, then s squared equals 4 pi. Since the side length of a square must be a positive number, taking the square root of both sides of s squared equals 4 pi gives s equals, the square root of 4 pi, end root. Using the properties of square roots, the square root of 4 pi, end root can be rewritten as open parenthesis, the square root of 4, close parenthesis, times, open parenthesis, the square root of pi, close parenthesis, which is equivalent to 2 times the square root of pi. Therefore, s equals, 2 times the square root of pi.

Choice A is incorrect. The side length of the square isn’t equal to the radius of the circle. Choices B and D are incorrect and may result from incorrectly simplifying the expression the square root of 4 pi, end root.

 

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