How to Calculate the Perimeter of a Square

The perimeter of a two-dimensional shape is the sum of the lengths of the border around the shape, or the sum of the lengths of its sides. ^{[1] X Research Source} By definition, a square is a 4-sided figure with equal side lengths, and 4 right angles (90°). ^{[2] X Research Source} Since all four sides are of equal length, finding the square’s perimeter is easy! This article shows you how to find the perimeter of a square if you know the length of one side. Then the article will show you how to find the perimeter of a square if only the area is known, and finally how to calculate the perimeter of the square inscribed in a circle with known radius.

Table of Contents

Calculate the perimeter when the length of a side is known

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Remember the formula for the perimeter of a square. For a square with side length S , the perimeter will be four times the length of that side: P=4s .

Calculate the perimeter when the area is known

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Formula for calculating the area of a square. The area of any rectangle (remember that a square is a special rectangle) is defined as the length times the width. ^{[3] X Research Source} Since the length and width of the square are equal, the area of the square with side length s will be s*s , or A = s ² .

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Take the square root of the area. Taking the square root of the area will tell you the length of one side of the square. Usually you have to use a calculator to get the square root, by entering the area value into the calculator and pressing the square root (√) key. You can also learn how to calculate square root by hand!

If the area of the square is 20, the side length s =√20 = 4,472 .
If the area of the square is 25 then s = √25 = 5 .

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Multiply the side length by 4 to find the perimeter. Use the side length s that you just calculated to substitute the formula for the perimeter P = 4s . The result is the perimeter of the square!

For a square with area 20 and side length 4,472, perimeter P = 4 * 4,472 = 17,888 .
For a square with area 25 and side length 5, perimeter P = 4 * 5 = 20 .

Find the perimeter of a square inscribed in a circle of known radius

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Understand what is an inscribed square. Inscribed shapes appear quite often on standardized tests like the GMAT and GRE, so you need to know the definition of incircle. An inscribed square is a square drawn within a circle such that its four vertices (angles) lie on the circle. ^{[4] X Research Sources}

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Identify the relationship between the radius of a circle and the length of the side of a square. The distance from the center of the inscribed square to each of its corners is equal to the radius of the circle. To find the length s , we have to imagine cutting the square in half diagonally, to get two right triangles. Each of these right triangles will have equal right-angled sides a and b , hypotenuse c , and it’s length is twice the radius of the circle, i.e. 2r .

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Use the Pythagorean Theorem to find the length of the side of the square. The Pythagorean theorem states that for any right triangle with sides a and b , and the hypotenuse c , then we have the formula a ² + b ² = c ² . ^{[5] X Research Source} Since sides a and b are equal (we’re still working with squares) and we know c = 2r , we can write down the equation and simplify it to find the side length as after:

a ² + a ² = (2r) ² “‘, now simplify the equation to:
2a ² = 4(r) ² , now divide both sides by 2:
(a ² ) = 2(r) ² , then take the square root of each side:
a = √(2r) . Side length s of the inscribed square = √(2r) .

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Multiply the side length of the square by 4 to find the perimeter. In this case, the perimeter of the square P = 4√(2r) . Because of the distributive property of the exponent, we know that 4√(2r) is equal to 4√2 * 4√r , simplifying this value for the following equation: the perimeter of any square inscribed in a circle has radius r will be P = 5,657r ! ^{[6] X Research Source}

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Solve an example equation. Given a square inscribed in a circle with radius 10. That is, the diagonal of this square = 2(10) = 20. Using the Pythagorean Theorem we have 2(a ² ) = 20 ² , so 2a ² = 400. Now divide both sides by 2 to get a ² = 200. Then take the square root of each side to get a = 14,142 . Multiply this by 4 and you will find the perimeter of the square: P = 56.57 .

Note that you can find the same result by multiplying the radius by 10 by 5,657. 10 * 5,567 = 56.57 , but it’s hard to remember this value while taking the test, so it’s better to remember the calculation process to find the result.

Advice

The reason is because the square has 4 sides.

Steps

Calculate the perimeter when the length of a side is known

Calculate the perimeter when the area is known

Find the perimeter of a square inscribed in a circle of known radius

Advice

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