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Geometry and Trigonometry Difficulty: Medium

What is the length of one side of a square that has the same area as a circle with radius 2 ?

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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 $the square root of 4 pi, end root$ .