Inscribed shapes come up fairly often on standardized tests like the GMAT and GRE, so it is important to know what they are. A square inscribed in a circle is a square that is drawn inside of the circle, so that all four vertices (corners) lie on the edge of the circle., The distance from the center of an inscribed square to each of its corners is equal to the radius of the circle. To find the length of s, we must first imagine cutting the square in half diagonally to form two right triangles. Each of these triangles will have equal sides a and b and hypotenuse c, which we know is equal to two times the radius of the circle, or 2r.
 The Pythagorean Theorem states that for any right triangle with sides a and b and hypotenuse c, a2 + b2 = c2 . Since sides a and b are equal (remember, we are still dealing with a square!) and we know that c = 2r, we can write out the equation and simplify the equation to find the side length as follows:


a2 + a2 = (2r)2"', now simplify the expressions:

2a2 = 4(r)2, now divide both sides by 2:

(a2) = 2(r)2, now take the square root of each side:

a = √(2r). Our side length s for the inscribed square = √(2r).
 In this case, the perimeter of the square P = 4√(2r). Because of the distributive properties of exponents, which tells us that 4√(2r) is equal to 4√2 * 4√r, we can simplify this to the following equation: the perimeter of any square inscribed in a circle with radius r is defined as P = 5.657r!, Consider a square inscribed in a circle with radius 10. That means that the diagonal of this square = 2(10), or 20. Using the Pythagorean Theorem, we know that 2(a2) = 202, so 2a2 = 400. Now divide both sides in half to find that a2 = 200. Then take the square root of each side to find that a = 14.142. Multiply this by 4, and you will find the perimeter of your square: P = 56.57.


Notice that you could have found the same thing by simply multiplying the radius, 10, by 5.657. 10 * 5.567 = 56.57, but that might be hard to remember on a test, so it's better to memorize the process we used to get there.
Understand what an inscribed square is.
Recognize the relationship between the radius of the circle, and the side length of the square.
Use the Pythagorean Theorem to find the side length of the square.
Multiply the side length of the square by four to find the perimeter.
Solve an example equation.
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