Explainer: Perimeters of Rectangles and Squares

In this explainer, we will learn how to find the perimeter of a rectangle or a square by adding the lengths of all their sides or by formula using whole numbers only.

You should already know the definition of the perimeter of a shape.

Definition: Perimeter

The perimeter of a shape is the total length of its boundary. It is calculated by adding together the length of all the sides.

For a square with side length π‘₯, the perimeter, which we can denote 𝑃, will be equal to the sum of the lengths of its sides: 𝑃=π‘₯+π‘₯+π‘₯+π‘₯.

Additionally, since all the sides are equal in length, the perimeter be calculated by multiplying the length of one of the sides by 4: 𝑃=4π‘₯.

This gives us a formula which describes how to find the perimeter of a square given its side length.

The Formula for the Perimeter of a Square

The perimeter 𝑃 of a square with side length π‘₯ can be calculated with the following formula: 𝑃=4π‘₯.

Now that we know the formula for the perimeter of a square, we will use it to solve some typical problems.

Example 1: Finding the Perimeter of a Square given Its Length

Find the perimeter of a square whose side length is 19 cm.

Answer

We are told that the length of each side of the square has length 19 cm.

So, to find the perimeter, 𝑃, we just have to add together the lengths of the 4 sides, 𝑃=19+19+19+19=76.

Alternatively, we can use the formula for the perimeter of a square which tells us that the perimeter is equal to 4π‘₯ where π‘₯ is the length of one of the sides. Hence, 𝑃=4Γ—19=76.

But, this is not the final answer. We have to make sure we include the correct length units when recording the perimeter of shapes. Since the length of the side was measured in centimeters, the perimeter we have calculated is also in centimeters. So, the perimeter of the square is 76 cm.

Example 2: Finding the Perimeter of a Square and Converting Units

Calculate, in cm, the perimeter of a square with side length 70 mm.

Answer

We know that the perimeter 𝑃 of a square can be calculated with the formula 𝑃=4π‘₯, where π‘₯ is the length of one side. We know that π‘₯=70 and so 𝑃=4Γ—70=280.

Since the length was given in millimeters, we have shown that the perimeter of the square is 280 mm.

Notice that the question asks us to calculate the length in centimeters. So, the final step is to convert 280 mm to centimeters. Recall that there are 10 mm in 1 cm, so to convert 280 millimeters into centimeters we have to divide by ten.

Hence, the perimeter of the square is 28010=28cm.

The above examples asked us to find the perimeter of a square given its length; a related question is to work backwards and find the side length of a square given its perimeter.

Since we know that 𝑃=4π‘₯, we can rearrange this formula by dividing both sides by 4 to obtain a formula to calculate the length π‘₯ given the perimeter 𝑃.

Formula Linking the Length and Perimeter of a Square

The side length π‘₯ of a square with perimeter 𝑃 can be calculated with the following formula: π‘₯=𝑃4.

Example 3: Finding the Length of a Square given Its Perimeter

Calculate the side length of a square in meters whose perimeter is 24 m.

Answer

The perimeter 𝑃 of the square is 24 m, and we have been asked to find the side length π‘₯.

The formulae linking the length and the perimeter of a square are 𝑃=4π‘₯andπ‘₯=𝑃4.

So, we calculate that π‘₯=244=6.

Finally, we must check that we use the correct units for our answer. The perimeter was given in meters and we were asked to find the length in meters so we conclude that the length is 6 m.

Example 4: Finding the Length of a Square given Its Perimeter

If a wire having a length of 96 cm is folded to form a square, how long will each side be?

Answer

The length of the wire is 96 cm and this is equal to the perimeter of the square that is created.

Since the 4 sides of the square have the same length, each side will be one-quarter of the length of the wire.

So, to find the length of a side we calculate 964=24.

Finally, we must include the correct units in the final answer. Since the wire was measured in centimeters, we have shown that the length of each side is 24 cm.

Example 5: Solving Problems about Perimeter of Squares

The sum of the perimeters of two squares is 108 mm. If the side length of one of them is 2 mm, find the side length of the other square in millimeters.

Answer

It can be helpful to sketch what we know to help us see how to answer the question. We know that there are two squares; let’s call them A and B.

One of the squares has side lengths equal to 2 mm; we will say that that is Square A. So, the question is asking us to find the side length of Square B. We will denote the unknown side length by π‘₯.

The last piece of information we are given is the sum of the perimeters, PerimeterofA+PerimeterofB=108mm, and we know that the perimeter of a square is equal to 4 times the side length.

Let’s write a formula for π‘₯ with this information.

Since the perimeter of Square A is equal to 8 mm and the perimeter of Square B is equal to 4π‘₯, we have the following equation for the sum of the perimeters: 8+4π‘₯=108.

We can solve this equation to find π‘₯.

First, subtract 8 from both sides: 4π‘₯=100.

Now, divide both sides by 4 to obtain π‘₯=25.

Finally, remember to include the correct units in your answer. The side length of Square B is 25 mm.

Finally, we will use what we know about the perimeter of squares to solve problems with more steps.

Here are the key steps to remember when solving problems with length and perimeter.

How to Solve Questions Involving Length and Perimeter of Squares

  1. If you know the side length π‘₯, you can multiply by 4 to obtain the perimeter 𝑃.
  2. If you know the perimeter, you can divide by 4 to obtain the side length.
  3. Remember to always include length units in your answer, and check whether the question wants you to use the same units or convert between units.

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