It is often going to be the case in the exam that you will be given the perimeter of a shape and the shape will have an unknown side. You will then be asked to work out what the length of the unknown side is. In this section, we are going to look at a few examples that are like this.

**Example 1**

What is the perimeter of the shape below? All of the angles in the shape below are right angles.

On the shape above we can see that the length of two of the sides are unknown. These two sides are labelled as

*x*and*y*. We are told in the question that all of the angles are right angles. This means that the length of the top of the shape is equal to 4 plus the unknown*x*. This gives us the following equation.We now solve like normal. We want to find the value of

*x*. This means that we need to get rid of the 4, which we do by taking 4 from both sides of the equation.*x*is equal to 6 cm.

We are now going to find the value of

*y*. 9 cm will be equal to be equal to 2 plus the unknown

*y*. This gives the equation below, which we can use to solve the value of

*y*.

We want to find the value of

*y*, so we take 2 from both sides of the equation.Therefore,

We are now able to work out the perimeter of the shape by adding all of the sides together. Starting from the bottom left corner, the calculation for the perimeter is (it does not matter where we start working out the perimeter from; all that matters is that we add all of the sides together):

*y*is equal to 7 cm.We are now able to work out the perimeter of the shape by adding all of the sides together. Starting from the bottom left corner, the calculation for the perimeter is (it does not matter where we start working out the perimeter from; all that matters is that we add all of the sides together):

The perimeter of the shape is 38 cm.

**Example 2**

Now let’s have a more complex example. The perimeter of the shape below is 34 m. What is the value of

*x*and work out the lengths of all of the sides for the shape?

All of the lengths in the following shape are unknown and in terms of

Therefore, we can create an equation whereby we set 34 m equal to all of the unknown sides added together. I am going to add all of the unknown sides up starting from the bottom left corner. This gives us the equation below:

*x*. We are told that the perimeter is 34 m and we know that the perimeter is worked out by adding all of the sides of the shape together.Therefore, we can create an equation whereby we set 34 m equal to all of the unknown sides added together. I am going to add all of the unknown sides up starting from the bottom left corner. This gives us the equation below:

We can now collect like terms on the right side of the equation; we collect all of the

*x*’s and all of the numbers.We now solve in the usual way. The first step is to isolate the

*x*terms; we need to get rid of the -11 that is currently on the right side of the equation. We are able to do this by doing the opposite; we add 11 to both sides of the equation.We want to find the value of

*x*and not 9*x*, so we divide both sides of the equation by 9.Therefore,

*x*is equal to 5. We are now able to work out the sides by subbing*x*as 5 into all of the lengths. The lengths of each of sides are shown in green on the diagram below.