GCSE Maths Geometry and Measure

Angles

Interior Angles of a Polygon

# Interior Angles of a Polygon

Here we will learn about interior angles in polygons including how to calculate the sum of interior angles for a polygon, single interior angles and use this knowledge to solve problems.

There are also angles in polygons worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

## What are interior angles?

Interior angles are the angles inside a shape. They are the angles within a polygon made by two sides:

Interior and exterior angles form a straight line – they add to 180°:

We can calculate the sum of the interior angles of a polygon by splitting it into triangles and multiplying the number of triangles by 180°.

E.g.

The number of triangles a polygon can be split into is always 2 less than the number of sides.

E.g.

A heptagon has 7 sides.

7-2=5, so we can split the heptagon into 5 triangles:

$5\times180^{\circ} = 900^{\circ}$

The general formula is:

Sum of Interior Angles = (n-2) × 180
‘n’ is the number of sides the polygon has

Step by step guide: Angles in polygons

### Keywords

• Polygon:
A polygon is a two dimensional shape with at least three sides, where the sides are all straight lines.
• Regular & irregular polygons:
A regular polygon is where all angles are equal size and all sides are equal length
E.g. a square
An irregular polygon is where all angles are not equal size and/or all sides are not equal length
E.g. a trapezium.

## How to solve problems involving interior angles

In order to solve problems involving interior angles:

1. Identify the number of sides in any polygon/s given in the question and note whether these are regular or irregular shapes.
2. Find the sum of interior angles for any polygon/s given.
3. Identify what the question is asking.
4. Solve the problem using the information you have already gathered.

## Interior angles examples

### Example 1: finding a single interior angle of a regular polygon

Find the size of each interior angle for a regular decagon.

1. Identify the number of sides in any polygon/s given in the question. Note whether this are regular or irregular shapes.

10 sides – regular shape.

2Find the sum of interior angles for any polygon/s given.

Sum of interior angles = (n-2) × 180

As a decagon has 10 sides:

n=10, so we can substitute n=10 into the formula.

Sum of interior angles of a decagon = (10-2) × 180

Sum of  interior angles of a decagon = 8 × 180

Sum of  interior angles of a decagon = 1440°

3Identify what the question is asking you to find.

The question is asking for ‘each interior angle’. This means the size of one interior angle.

4Solve the problem using the information you have already gathered.

We know the sum of the interior angles for this polygon is 1440°.

We know, as it is a regular polygon, that all the angles are of equal size.

Therefore we can find the size of each interior angle by dividing the sum of interior angles by the number of angles in the polygon:

$\text{Each interior angle} = \frac{1440}{10}$

The size of each interior angle is 144°.

Step by step guide: Substitution

### Example 2: finding a single interior angle of an irregular polygon

The diagram shows a polygon. Find the size of angle x.

6 sides – irregular hexagon

Sum of interior angles  = (n-2) × 180

Sum of interior angles for a hexagon = (6-2) × 180

Sum of interior angles for a hexagon = 720°

Finding the missing angle labelled as x.

Note that we know the values of all the other angles.

\begin{aligned} 120+90+101+130+160+x&=720 \\ 601+x&=720 \\ x&=119 \end{aligned}

The size of angle is 119°.

### Example 3: finding the number of sides given the interior angle of a regular polygon

Each of the interior angles of a regular polygon is 140°. How many sides does the polygon have?

Unknown number of sides – regular shape

Sum of interior angles  = (n-2) × 180

We need to find the number of sides.

We know a single angle of this regular polygon is 140°.

Therefore all the angles are 140°.

We can write the sum of the interior angles as 140 multiplied by the number of sides or 140n.

Therefore:

\begin{aligned} 140 n&=(n-2) \times 180 \\ 140 n&=180 n-360 \\ 360&=40 n \\ 9&=n \end{aligned}

The polygon has 9 sides.

NOTE: We can also solve this problem by calculating an exterior angle.

Step by step guide: exterior angles of a polygon

### Example 4: multiple shapes

Shown below are three congruent regular pentagons. Find angle y.

Each polygon has 5 sides (pentagon) and is regular.

As each polygon shown is a regular pentagon they all have equal sums of their interior angles:

Sum of interior angles = (n-2) × 180

Sum of interior angles for a pentagon = (5-2) × 180

Sum of interior angles for each pentagon = 540°

Find the missing angle y shown on the diagram.

We know that angles around a point add to 360°, so if we add the three interior angles shown and y together we will get 360°.

Each interior angle shown is 540 ÷ 5 = 108°

We can now calculate y by forming an equation:

\begin{aligned} 108+108+108+y&=360 \\ 324+y&=360 \\ y&=36 \end{aligned}

Angle y is equal to 36°.

### Example 5: problem solving to find the number of sides

Shown below are sections of three identical regular polygons where AB, BC and CA are all sides of the polygons.

ABC is an equilateral triangle formed by placing the three larger polygons together.

Calculate the number of sides each regular polygon has.

Shown is an equilateral triangle (regular shape) made up of the adjacent sides AB, BC and CA.

We need to calculate the number of sides of the larger polygons.

An equilateral triangles has the sum of interior angles of 180°.

We do not know the number of sides of the polygons so their sum of interior angles can be represented by (n-2) × 180.

The number of sides of the regular polygons where we are only shown one side.

Looking at point A we can see there are three angles around a point. One of the angles is within the equilateral triangle, so it must be 60°, and the other two angles are from the polygons we are attempting to find.

We will call these angles x:

We know that angles around a point add to 360°.

Therefore:

\begin{aligned} 60+2 x &=360 \\ 2 x &=300 \\ x &=150 \end{aligned}

This means that each interior angle of the regular polygon is 150°.

So the sum of interior angles is equal to 150 × n or 150n

150n = (n-2) × 180

We can now solve for n:

\begin{aligned} 150 n&=(n-2) \times 180 \\ 150 n&=180 n-360 \\ 360&=30 n \\ 12&=n \end{aligned}

The polygon has 12 sides, so each polygon shown in the diagram has 12 sides.

### Example 6: problem solving to find the number of angles

Shown is a regular pentagon. Find y.

5 sides – regular

Sum of interior angles = (n-2) × 180

Sum of interior angles for a decagon = (5-2) × 180

Sum of interior angles for a decagon = 540°

Find angle y which is within one of the interior angles.

As the polygon is regular you can find the size of one interior angle by:

540° ÷ 5 = 108

As the polygon is regular AC = AB

Therefore ABC is an isosceles triangle where angles ACB and ABC are equal to one another and are therefore both y.

We know that the interior angles of a triangle add to 180°

Therefore,

\begin{aligned} 108+y+y&=180 \\ 108+2 y&=180 \\ 2 y&=72 \\ y&=36 \end{aligned}

Angle y is equal to 36°.

### Common misconceptions

• Miscounting the number of sides
• Misidentifying if a polygon is regular or irregular
• Dividing the sum of interior angles by the number of triangles created.
You should divide by the number of sides to find the size of one interior angle (for regular polygons only)
• Incorrectly assuming all the angles are the same size
• Misidentifying which angle the questions is asking you to calculate

### Practice interior angles of a polygon questions

1. Find the sum of interior angles for a polygon with 13 sides

2340^{\circ}

1980^{\circ}

4680^{\circ}

3960^{\circ}

Sum of Interior Angles = (n-2)\times180

In this case n=13 , so the calculation becomes 11 \times 180 .

2. Find the size of one interior angle for a regular quadrilateral

360^{\circ}

180^{\circ}

72^{\circ}

90^{\circ}

The sum of interior angles in a quadrilateral is 360^{\circ} . For a regular shape all the angles are the same size, so we divide 360 by 4 to arrive at the answer.

3. Find the size of one interior angle for a regular nonagon

360^{\circ}

140^{\circ}

180^{\circ}

135^{\circ}

The sum of interior angles in a nonagon is 1260^{\circ} . For a regular shape all the angles are the same size, so we divide 1260 by 9 to arrive at the answer.

4. Each of the interior angles of a regular polygon is 165^{\circ} . How many sides does the polygon have?

 20  sides

 22  sides

 24  sides

 26  sides

Sum of Interior Angles = (n-2)\times180

With this in mind, we have 165n=(n-2) \times 180

Which simplifies to 15n = 360

So n=24

5. Each of the interior angles of a regular polygon is 160^{\circ} . How many sides does the polygon have?

 16  sides

 18  sides

 20  sides

 22  sides

Sum of Interior Angles = (n-2)\times180

With this in mind, we have 160n=(n-2)\times180

Which simplifies to 20n = 360

So n=18

6. Four interior angles in a pentagon are each 115^{\circ} . Find the size of the other angle.

540^{\circ}

80^{\circ}

460^{\circ}

305^{\circ}

By using the formula,

Sum of Interior Angles = (n-2)\times180

We know that a pentagon has interior angles that add up to 540^{\circ} .

### Interior angles of a polygon GCSE questions

1. Work out the size of the angle labeled x .

(3 marks)

(6-2) \times 180 = 720

(1)

80 + 55 + 280 + 25 + 162 = 602

(1)

720-602=118

(1)

`2. The diagram below shows a regular decagon.

(a) Work out the size of angle a .

(b) Work out the size of angle b.

(5 marks)

(10-2) \times 180 = 1440

(1)

1440 \div 10 = 144

(1)

144\times 2 = 288

(1)

360 – 288 = 72

(1)

72 \div 2=36 (1)

3. A regular polygon’s interior and exterior angles are in the ratio 9 : 1 . How many sides does the polygon have?

(4 marks)

180^{\circ} in ratio 9:1

180 \div 10=18, 18 \times 9=162, 18 \times 1= 18

(1)

One interior angle =162^{\circ}

(1)

\begin{aligned} 162n&=(n-2) \times 180 \\ 162n&=180n-360 \end{aligned}

18n=360

(1)

n=20

(1)

## Learning checklist

You have now learned how to:

• Use conventional terms for geometry e.g. interior angle
• Knowing names and properties of polygons
• Calculate the sum of interior angles for a regular polygon
• Derive and use the sum of angles in a triangle to deduce and use the angle sum in any polygon, and to derive properties of regular polygons
• Calculate the size of the interior angle of a regular polygon

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