Polygons – sum of interior angles

## Key points

• A polygon is a closed 2D shape with at least three sides.

• A regular polygon is a polygon with all equal sides and angles.
• An irregular polygon is a polygon where this is not the case.
• Knowing the difference between regular and irregular polygons can help when working out the size of missing angles.
• In order to work out the size of missing interior angles in polygons, it is important to know what the interior angles add up to:

• The interior angles of a triangle sum to 180°.
• The interior angles of a sum to 360°.
• The interior angles of a sum to 540°.
• In general, the interior angles of any polygon sum to (number of sides – 2) × 180°.
• To find the size of one interior angle of a regular polygon, divide the sum of the interior angles by the number of sides.

• To find the size of a missing interior angle in an irregular polygon, subtract the sum of the given angles from the sum of the interior angles.

## Finding the sum of interior angles in polygons

• The angles inside a shape are known as .

• The interior angles in any triangle sum to 180°.

• Any can be divided into two triangles (less than the number of sides of the polygon).

• The sum of the interior angles of any polygon can be found using the formula:

sum of interior angles = (number of sides – 2) × 180°

• It can be useful to remember some key polygons and the sum of their interior angles:
 Polygon Number of sides Formula Sum of interior angles triangle 3 (3 – 2) × 180 180° quadrilateral 4 (4 – 2) × 180 360° pentagon 5 (5 – 2) × 180 540° hexagon 6 (6 – 2) × 180 720° \(n\)-sided polygon \(n\) (\(n\) – 2) × 180 (\(n\) – 2) × 180°

### Examples

The reason interior angles in a quadrilateral sum to 360° is because a quadrilateral can be divided into two triangles. The interior angles in each triangle sum to 180° so two triangles together sum to 360°.

A pentagon can be divided into three triangles. Interior angles in a triangle sum to 180°. 180 × 3 = 540. Interior angles in a pentagon therefore sum to 540°.

A hexagon can be divided into four triangles, therefore the sum of the interior angles of a hexagon is 180 × 4 = 720. The interior angles in a hexagon sum to 720°.

Continuing this process generates the pattern that the number of triangles a polygon can be divided into is always two less than the number of sides the polygon has. This can be written as (𝒏 – 2), where 𝒏 stands for number of sides.

This is an irregular decagon. To work out what the interior angles in a decagon sum to, there are two methods.

The first method is to work out how many triangles we can divide the decagon into, and then multiply this by 180°. This gives 8 × 180 = 1440. The interior angles in a decagon sum to 1440°. The second method is to use the formula: sum of interior angles = (𝒏 – 2) × 180. This gives (10 – 2) × 180 = 1440

Find the sum of the interior angles of a 26-sided polygon. It would not be practical to draw a 26-sided polygon, so the formula is more useful here. (26 – 2) × 180 = 24 × 180 = 4320. The interior angles of a 26-sided polygon sum to 4320°.

Find the sum of interior angles of a 53-sided polygon. It would not be practical to draw a 53-sided polygon, so the formula is more useful here. (53 – 2) × 180 = 51 × 180 = 9180. The interior angles of a 53-sided polygon sum to 9180°.

Find the number of sides when the sum of the interior angles is known, 1620°. Solve the equation (𝒏 – 2) × 180 = 1620. Divide both sides of the equation by 180, then add 2 to both sides. This shows that the number of sides the polygon has is 11

Find the number of sides when the sum of the interior angles is known, 2160°. Solve the equation (𝒏 – 20) × 180 = 2160. Divide both sides of the equation by 180, then add 2 to both sides. This shows that the number of sides the polygon has is 14

### Question

What do the interior angles of this irregular octagon sum to?

The interior angles in an octagon sum to 1080°.

Using the formula:

sum of interior angles = (\(n\) – 2) × 180°

= (8 – 2) × 180 = 6 × 180 = 1080

## Finding missing interior angles in polygons

• A regular polygon is a polygon with all sides and angles equal.

• on each side indicate which sides are equal. If all sides are marked with a single hash mark, the polygon is regular.
• If the sum of the is known, dividing by the number of sides will give the size of one angle.
• An irregular polygon is a polygon with all sides and angles not equal.

• It is not possible to find the size of one angle in an irregular polygon unless all other angles are known.

### Examples

These polygons are all pentagons. Only the one on the left is regular. This is indicated by the hash marks on each side showing that they are equal.

Find the size of one interior angle of a regular pentagon. In order to do this, the sum of the interior angles must first be known. Use the formula, (𝒏 – 2) × 180, where 𝒏 is the number of sides of the polygon, which gives (5 – 2) × 180 = 3 × 180 = 540. The sum of the interior angles in a pentagon is 540°.

A regular pentagon has five angles and all of them are equal. To find the size of one angle, divide 540 by 5 to get 108. Each interior angle is equal to 108°.

Find the size of each interior angle in a regular octagon. First, find the sum of the interior angles using the formula: (𝒏 – 2) × 180 = (8 – 2) × 180 = 6 × 180 = 1080. Then divide this by the number of sides, eight: 1080 divided by 8 = 135. Each interior angle is equal to 135°.

Find the size of angle 𝒙 in this irregular heptagon. In this case, the size of angle 𝒙 cannot be found as the other angles are not known.

Find the size of angle 𝒙 in this irregular heptagon. The sum of the interior angles in a heptagon is (7 – 2) × 180 = 5 × 180 = 900°. The known angles add up to 150 + 155 + 128 + 89 + 110 + 166 = 798. To find the final missing angle, 𝒙, subtract this from 900. 900 – 798 = 102. Angle 𝒙 is 102°.

Find the final missing angle, 𝒎, in this irregular nonagon. The sum of the interior angles in a nonagon is (9 – 2) × 180 = 7 × 180 = 1260°. The known angles add up to 96 + 100 + 190 + 140 + 113 + 127 + 155 + 122 = 1043. To find the final missing angle, 𝒎, subtract this from 1260. 1260 – 1043 = 217. Angle 𝒎 is 217°.

Find the missing angle, 𝒂, in this irregular hexagon. The sum of the interior angles in a hexagon is (6 – 2) × 180 = 4 × 180 = 720°. The known angles add up to 104 + 95 + 107 + 84 = 390. This means that 3𝒂 must be 720 – 390 = 330. 330 divided by 3 = 110, so angle 𝒂 = 110°.

### Question

Find the size of angle \(z\).

This is an irregular octagon.

The sum of the interior angles in an octagon is (8 – 2) × 180 = 6 × 180 = 1080°.

The known angles add up to 240 + 105 + 136 + 148 + 154 + 127 + 100 = 1010

To find the final missing angle, \(z\), subtract 1010 from 1080

1080 – 1010 = 70

Angle \(z\) is 70°.

## Practise the sum of interior angles of polygons

Practise finding the size of interior angles of polygons with this quiz. You may need a pen and paper to help you with your answers.

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