# What is the sum of all the angles in a 4 sided shape (angles in a quadrilateral add up to 360 degrees)

Angles in a quadrilateral add up to 360⁰. This is because you can divide up a quadrilateral into 2 triangles and since the angles in a triangle add up to 180⁰, then 2 × 180⁰ = 360⁰. So if you are asked to find a missing angle in a quadrilateral, find the total of the 3 angles that are already given and subtract this total off 360⁰.

Example 1

Work out angle x in this quadrilateral.

First of all work out the total of the 3 angles that are given:

44 + 124 + 65 = 233⁰

All you need to do now is take this total off 360⁰

360 – 233 = 127⁰.

Example 2

Work out angle y in this trapezium.

Again, find the total of the 3 angles that are given. Also, make sure you count the 2 right angles that are 90⁰ each. So:

126 + 90 + 90 = 306⁰

All you need to do next is take this total off 360⁰

360 – 306 = 54⁰.

Example 3

Four angles in a quadrilateral have angles equal to 3x + 6, 2x + 3, x + 11 and 4x. Calculate the size of all of the angles in this quadrilateral.

Let’s begin by adding up the 4 angles that are given. This will give us another expression.

3x + 6 + 2x+ 3 + x + 11 + 4x

Now simplify this expression (add the x terms together and the numbers together)

= 10x + 20

As shown above the angles in a 4 sided shape add up to 360⁰, so you can make an equation involving x:

10x + 20 = 360.

If you know solve this equation you will be able to find the value of x.

10x + 20 = 360 (take 20 off both sides)

10x = 340 (divide both sides by 10)

x = 34⁰.

Finally, if you substitute x = 34 into the 4 expressions above this will give the size of all the angles.

3x + 6 = 3 × 34 + 6 = 108⁰

2x + 3 = 2 × 34 + 3 = 71⁰

x + 11 = 34 + 11 = 45⁰

4x = 4 × 34 = 136⁰

You can check that these 4 angles are the correct size by seeing if they sum to 360⁰.

108 + 71 + 45 + 136 = 360⁰.

So these 4 angles are the correct size.