The perimeter of any polygon is simply the combined lengths of all its sides. In a polygon like a rectangle or a parallelogram, where the two opposite sides are equal, this can be simply represented as P = length + height + length + height = 2·(length + height)

A common geometry problem using the perimeter of a rectangle involves providing the perimeter and one of the sides and asking you to find the other side. This can be done using simple algebra.

Here are a couple of examples.

## Problem 1

A rectangle has a perimeter of 100 inches. The longer side is 20 inches longer than the shorter one. Find the length of each side.

### Solution

Call the shorter side x. The longer side is x=20. Then:

(1) 100 = 2·(length + height) = 2·(x+x+20)

(2) 50 = 2x+20

(3) 30 = 2x

(4) x=15

So the short side is 15 inches, and the long side is 35 inches.

A slightly more complex problem is the following:

## Problem 2

Given a rope of length 100 inches, you can create rectangles with different perimeters. For example: a rectangle with sides measuring 10,10, 40 & 40; or one with sides measuring 20,20,30 & 30. Each rectangle has a different area (the first one in the example has an area of 10×40 or 400 square units; the second one is 20×30=600). What is the rectangle with the largest area you can create with that rope?

### Solution

These types of maximization or minimization problems are easily solved using basic calculus techniques – finding the derivative of a function and solving for zero. But this problem is simple enough to solve without requiring any knowledge of calculus.

The area of a rectangle is (length x height). If we have a fixed perimeter of 100, and the perimeter is 2·(length + height), then (length + height) =50.

If we call one side x, the two sides can be represented as x and (50-x). The area of any rectangle we make form this rope is then A=x·(50-x), or A=50x-x^{2}.

From our knowledge of quadratic functions, we know that A=50x-x^{2 }is a parabola with a maximum occurring at x=-50/-2=25, so the largest rectangle is one in which the sides are 25 and 50-25=25 – a square! That square has an area of 625 square units.