The square is a geometric shape that needs no introduction. It's a rectangle, which means that it has four sides and four 90-degree angles, but it's a special case of this two-dimensional shape. All of its four sides are equal. This fact makes it especially easy to calculate the length of one of the sides, given the area of the square. If the area enclosed by the square is A, and the length of each side is L, then L = √A. You might have an opportunity to make use of this simple conversion if you're planning to construct a fence around a square parcel of land with a known acreage.

#### TL;DR (Too Long; Didn't Read)

The area of a square with sides of length L X L, or L^{2}. Since A = L^{2}, it follows that L = √A.

## Deriving the Relationship Between Area and Side Length

Many geometric shapes have four sides, but to be a rectangle, the shape must have four right angles. Because of this requirement, a rectangle can have sides of two different lengths, but no more. For example, a tapered figure with two sides of equal length and two ends of different lengths is not a rectangle.

Considering a rectangle with sides of lengths L and W, basic geometry tells you that its area (A) is LW.

A = LW

In other words, you find the area by multiplying the length of the rectangle by the width. The same is true for a square, but there's a key difference: For a square, the length and width are equal. If the length is L, then the area of the square is L^{2}.

A = L^{2}

If you know the area of the square, you can immediately calculate the length of each of its sides by rearranging the above equation:

L = √A.

## A Real-World Application

A farmer has a square plot of land with an area of 3 acres. If he wants to fence the land to make a horse corral, how much fencing does he need?

## Convert Acres to Square Feet

There are 43,560 square feet in an acre, so the area of the farmer's land is 3 • 43,560 = 130,680 square feet.

## Find the Square Root of the Area

Finding the square root is easier if you convert large numbers to scientific notation. Accordingly, 130,560 = 1.3056 X 10^{5} sq. ft. The square root is 361.33 feet. This is the length (L) of one side of the plot of land.

## Calculate the Perimeter of the Square

The perimeter is the total distance around the square. For a rectangle, the perimeter is 2(L +W). For a square, which has four equal sides, the perimeter is 4L. In the farmer's case, the perimeter is 1,445.32 feet. To make sure he has enough materials, the farmer should probably buy enough for 1450 feet of fencing.