Angles In Polygons

Here we will learn about angles in polygons including how to calculate angles in polygons using a variety of methods and an overview of interior and exterior angles

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 angles in polygons?

Angles in polygons relate to the interior and exterior angles of regular and irregular polygons.

Interior angles are the angles within a polygon made by two sides.
We can calculate the sum of the interior angles of a polygon by subtracting 2 from the number of sides and then multiplying by 180º.

Step-by-step guide: Interior angles of a polygon

Exterior angles are the angles between a polygon and the extended line from the next side.
The sum of the exterior angles of a polygon is always equal to 360º.

Step-by-step guide: Exterior angles of a polygon

A polygon is a two dimensional shape with at least three sides, where the sides are all straight lines.
‘Poly’ comes from the greek for ‘many’ whilst ‘gon’ means ‘angles’.
You will be familiar with many types of polygons such as triangle, rectangle and pentagon.

Regular polygons have all angles are that are equal in size and all sides that are equal in length.

Irregular polygons have angles that are not equal in size and sides that are not equal in length.

Sum of the interior angles of a polygon

The ‘sum of interior angles’ of a polygon means finding the total of all the angles in a polygon. This is the key step in helping us solve many problems involving angles in polygons.

We know that the sum of all the angles in a triangle is equal to 180º, but what about angles in a quadrilateral? A regular pentagon? Or even an irregular octagon?

Sum of the angles in a triangle:

We know that the three angles in any triangle add up to 180º.

Mathematically we would say:

“The sum of interior angles for a triangle is 180 degrees”.

Step-by-step guide: Angles in a triangle

Sum of the angles in a quadrilateral:

A quadrilateral is a four sided shape. We can ‘split’ a quadrilateral into two triangles by drawing a line from one corner to an opposite one.

If the sum of interior angles one triangle is 180º , then the sum of the interior angles of two triangles is 180º × 2 = 360º.

So the sum of the interior angles of quadrilateral is 360º.

Step-by-step guide: Angles in a quadrilateral

Using our knowledge of triangles we can find the sum of the interior angles of any polygon by splitting it into triangles.

Step-by-step guide: Angles in a pentagon

Step-by-step guide: Angles in a hexagon

Key relationships

Interior and exterior angles add up to 1800.

The number of triangles created inside a shape is always 2 lower than the number of sides.

The sum of the interior angles of a polygon depends on how many sides it has, not what it looks like.

How to find the sum of the interior angles of a polygon

In order to find the sum of interior angles for any polygon you should:

1. Identify how many sides the polygon has.
2. Identify if the polygon is regular or irregular.
3. If possible work out how many triangles could be created within the polygon by drawing lines from one vertex to all the other vertices.
4. Multiply the number of triangles by 180 to calculate the sum of the interior angles.
5. State your findings e.g. sides, regular/irregular, the sum of interior angles.

Angles in polygons examples

Find the sum of interior angles for this polygon.

1. Identify how many sides the polygon has.

This polygon has four sides.

2 Identify if the polygon is regular or irregular.

This polygon is irregular as all the sides are not of equal length.

3 If possible work out how many triangles could be created within the polygon by drawing lines from one corner to all the other vertices.

The polygon can be broken up into two triangles.

4 Multiply the number of triangles by 180 to get the sum of the interior angles.

$180^{\circ} × 2 = 360^{\circ}$

5 State your findings e.g. sides, regular/irregular, the sum of interior angles.

The polygon is a irregular quadrilateral (specifically called a parallelogram as both opposite sides are parallel) with a sum of interior angles of 360º.

Example 2: regular decagon

Identify how many sides the polygon has.

Identify if the polygon is regular or irregular.

If possible work out how many triangles could be created within the polygon by drawing lines from one corner to all the other vertices.

Multiply the number of triangles by 180o to get the sum of the interior angles.

State your findings e.g. sides, regular/irregular, the sum of interior angles.

Example 3: irregular pentagon with concave angle

Identify how many sides the polygon has.

Identify if the polygon is regular or irregular.

If possible work out how many triangles could be created within the polygon by drawing lines from one corner to all the other vertices.

Multiply the number of triangles by 180o to get the sum of the interior angles.

State your findings e.g. sides, regular/irregular, the sum of interior angles.

Example 4: complex polygon

Identify how many sides the polygon has.

Identify if the polygon is regular or irregular.

If possible work out how many triangles could be created within the polygon by drawing lines from one corner to all the other vertices.

Multiply the number of triangles by 180o to get the sum of the interior angles.

State your findings e.g. sides, regular/irregular, the sum of interior angles.

Example 5: finding an exterior angle

What is the size of one of the exterior angles of an equilateral triangle?

Example 6: finding the sum of interior angles of larger polygons

What is the sum of the interior angles for a 30 sided polygon?

Example 7: finding the sum of interior angles of a complex polygon

What is the sum of interior angles of this polygon?

Common misconceptions

• Miscounting the number of sides

• Misidentifying if a polygon is regular or irregular

• Incorrectly assuming all the angles are the same size

• Crossing lines when drawing the triangles, this creates false interior angles

Practice angles in polygons questions

1. Is a square a regular or irregular polygon?

Regular

Irregular

All the side lengths are the same, and all angles are right angles, hence a square is a regular polygon.

2. Is a semi circle a polygon?

Yes

No

A semicircle has a side which is not straight, so it is not a polygon.

3. What is the sum of interior angles for a triangle?

90^{\circ}

180^{\circ}

360^{\circ}

270^{\circ}

The angles in a triangle add up to 180^{\circ}.

4. What is the sum of interior angles for a regular hexagon?

720^{\circ}

540^{\circ}

360^{\circ}

1080^{\circ}

We know the angles in a triangle add up to 180^{\circ} . A regular hexagon can be divided into 4 triangles, and four lots of 180 is 720 .

5. What is the sum of interior angles for an irregular hexagon?

720^{\circ}

540^{\circ}

360^{\circ}

1080^{\circ}

We know the angles in a triangle add up to 180^{\circ} . An irregular hexagon can be divided into 4 triangles, and four lots of 180 is 720 .

6. What is the sum of interior angles for a regular 12 sided polygon?

2700^{\circ}

2160^{\circ}

1800^{\circ}

1080^{\circ}

We know the angles in a triangle add up to 180^{\circ} . A 12 sided polygon can be divided into 10 triangles, and ten lots of 180 is 1800 .

7. What is the sum of interior angles for a regular 25 sided polygon?

4500^{\circ}

4140^{\circ}

9000^{\circ}

3600^{\circ}

We know the angles in a triangle add up to 180^{\circ} . A 25 sided polygon can be divided into 23 triangles, and 23 lots of 180 is 4140 .

Angles in polygons GCSE questions

1.  Each exterior angle of a regular polygon is 15^{\circ} .
Work out the number of sides the polygon has.

(2 marks)

360 \div 15

(1)

24

(1)

2.  Each of the interior angles of a regular polygon is 140^{\circ} . Show that this polygon has 9 sides

(2 marks)

Exterior angle = 40 seen or implied

(1)

360 \div 40=9

(1)

3.  In a regular polygon each exterior angle is 18^{\circ} . Find the sum of interior angles for this polygon

(3 marks)

360 \div 18 \quad \text { or implied by "20" }

(1)

180 \times 18

(1)

3240

(1)

Learning checklist

You have now learned how to:

• Use conventional terms for geometry e.g. regular or irregular
• Derive and apply the properties and definitions of: special types of quadrilaterals
• Knowing names and properties of polygons
• Calculate the sum of interior angles for a regular polygon
• Calculate the sum of interior angles for a irregular polygon

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