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2D shapes Angles on a straight line Angles in a triangle Angles in polygonsThis topic is relevant for:

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.

**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

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

\[5\times180^{\circ} = 900^{\circ}\]

The general formula is:

**Sum of Interior Angles = (n – 2) × 180°**

**Step by step guide:** Angles in polygons

**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.

In order to solve problems involving interior angles:

**Identify the number of sides in any polygon/s given in the question.****Note whether the shape is regular or irregular.****Find the sum of interior angles for any polygon/s given.****Identify what the question is asking.****Solve the problem using the information you have already gathered****with use of the formulae**\textbf{+}**interior angle**\, \textbf{=} \; \bf{180^{\circ}}*exterior angle***and**\, \textbf{=} \, \bf{360^{\circ}}**Sum of exterior angles****if required**

Get your free interior and exterior angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREEGet your free interior and exterior angles worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE**Interior angles of a polygon** is part of our series of lessons to support revision on **angles in polygons**. You may find it helpful to start with the main angles in polygons lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

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

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

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

Sum of interior angles =

As a decagon has

Sum of interior angles of a decagon =

Sum of interior angles of a decagon =

Sum of interior angles of a decagon =

**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 with use of the formulae *** interior angle* \textbf{+}

We know the sum of the interior angles for this polygon is

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

**Step-by-step guide: **Substitution

The diagram shows a polygon. Find the size of angle

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

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

Sum of interior angles =

Sum of interior angles for a hexagon =

Sum of interior angles for a hexagon =

**Identify what the question is asking you to find.**

Finding the missing angle labelled as

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

**Solve the problem using the information you have already gathered** **with use of the formulae** * interior angle* \textbf{+}

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

The size of angle is

Each of the interior angles of a regular polygon is

Unknown number of sides – regular shape

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

Sum of interior angles =

**Identify what the question is asking you to find.**

We need to find the number of sides.

**Solve the problem using the information you have already gathered** **with use of the formulae** * interior angle* \textbf{+}

We know a single angle of this regular polygon is

Therefore all the angles are

We can write the sum of the interior angles as

Therefore:

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

The polygon has

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

Shown below are three congruent regular pentagons. Find angle

Each polygon has

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

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

Sum of interior angles =

Sum of interior angles for a pentagon =

Sum of interior angles for each pentagon =

**Identify what the question is asking you to find.**

Find the missing angle

**Solve the problem using the information you have already gathered** **with use of the formulae** * interior angle* \textbf{+}

We know that angles around a point add to

Each interior angle shown is

We can now calculate

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

Angle

Shown below are sections of three identical regular polygons where

Calculate the number of sides each regular polygon has.

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

Shown is an equilateral triangle (regular shape) made up of the adjacent sides

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

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

An equilateral triangles has the sum of interior angles of

We do not know the number of sides of the polygons so their sum of interior angles can be represented by

**Identify what the question is asking you to find.**

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

**Solve the problem using the information you have already gathered** **with use of the formulae** * interior angle* \textbf{+}

Looking at point

We will call these angles

We know that angles around a point add to

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

So the sum of interior angles is equal to

We can now solve for

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

The polygon has

Shown is a regular pentagon. Find

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

Sum of interior angles =

Sum of interior angles for a decagon =

Sum of interior angles for a decagon =

**Identify what the question is asking you to find.**

Find angle

**Solve the problem using the information you have already gathered** **with use of the formulae** * interior angle* \textbf{+}

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

As the polygon is regular

Therefore **isosceles **triangle where angles

We know that the interior angles of a triangle add to

Therefore,

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

Angle

**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**

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} .

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

**(3 marks)**

Show answer

(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)**

Show answer

(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)**

Show answer

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)**

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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