KS3

Angles in polygons

Part ofMathsAngles

Key points

An image showing two shapes, a regular nonagon and a regular decagon. Written below the shapes: nonagon, decagon. — Image caption,
A nonagon has nine sides and a decagon has ten sides.

To understand polygons, a good understanding of other angle rules, including triangles and quadrilaterals can be helpful.
A is a with three or more sides.
In a polygon, an interior and exterior angle add up to 180°.

An image showing two shapes, a regular nonagon and a regular decagon. Written below the shapes: nonagon, decagon. — Image caption,
A nonagon has nine sides and a decagon has ten sides.

What is a polygon?

Triangles, quadrilaterals and pentagons are all examples of polygons.

A polygon is a two-dimensional closed shape with straight sides.
The sides of a polygon are called .
Polygons have special names depending on the number of sides.

Example

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A regular polygon has sides (edges) that are all the same length and interior angles that are all equal.
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An irregular polygon has sides that are all different lengths and its interior angles are not all equal.
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This shape is called a hexagon because it has six edges and six interior angles.
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All the shape’s edges are the same length.
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All the shape’s angles are equal. This means the shape is regular.
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Each polygon has a name based on its number of sides. A pentagon has five sides, a heptagon has seven sides and an octagon has eight sides.
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A nonagon has nine sides and a decagon has ten sides.
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The word for the name of the shape includes a prefix that indicates the number of sides and the suffix ‘-gon’. For example, a twenty-sided shape is called an icosagon. The prefix ‘icosa’ is ancient Greek for twenty. The suffix ‘-gon’ is ancient Greek for side or angle.

Question

Which of these shapes are regular?

An image showing four shapes, a square, an irregular quadrilateral, an irregular pentagon, and a regular decagon.

What are interior and exterior angles?

In a polygon:

An interior angle is an angle inside the shape.
An exterior angle is created by extending an edge.
An interior angle and an exterior angle add up to 180°.

Examples

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This is an irregular pentagon.
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The interior angles are inside the shape.
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An exterior angle is created by extending an edge.
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Notice that the adjacent interior and exterior angles make a straight line. This means they must add up to 180°. Knowing the size of the interior angle makes it possible to calculate the exterior angle. Exterior angle = 180° – interior angle.
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An equilateral triangle has three equal sides. Its angles are also equal and each measures 60°.
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By extending the horizontal base an exterior angle is created. This means an exterior angle for an equilateral triangle must be 120°, since 180 – 60 = 120
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A regular octagon has an exterior angle of 45°.
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This means an interior angle for a regular octagon must be 135°.
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An equilateral triangle has three sides. The interior angle is 60° and the exterior angle is 120°. A square has four sides. The interior angle is 90° and the exterior angle is 90°. A regular pentagon has five sides. The interior angle is 108° and the exterior angle is 72°.
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A regular hexagon has six sides. The interior angle is 120° and the exterior angle is 60°. A regular octagon has eight sides. The interior angle is 135° and the exterior angle is 45°. A regular nonagon has nine sides. The interior angle is 140° and the exterior angle is 40°.

Question

A 50p coin is an example of a heptagon (7 edges).

If the interior angle of a regular heptagon is 129° (to the nearest degree), what is the size of an exterior angle?

An image of two fifty pence coins held in a hand.

Properties of exterior angles

For any , the exterior angles will always add up to 360°.
For a polygon, the size of one exterior angle = \( \frac{360}{n} \), where \(n\) is the number of sides.

Examples

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This polygon is a scalene triangle. It is irregular, and all its sides and angles are different.
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All the angles in a triangle sum 180°. 85 + 80 + 15 = 180°.
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The exterior angle for each vertex (corner) can be calculated by extending the edges of the triangle.
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The interior and exterior angles add up to 180. Subtract the interior angles from 180° to find the exterior angles. For the exterior angle at A, 180 – 80 = 100°. For the exterior angle at B, 180 – 85 = 95°. For the exterior angle at C, 180 – 15 = 165°.
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These exterior angles add up to 360°. This is true for all polygons. 100 + 95 + 185 = 360°.
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In a regular pentagon, all the interior angles are the same size and all the exterior angles are the same size.
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The five exterior angles add up to 360°, and each angle is equal.
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The exterior angle is found by dividing 360° by the number of exterior angles. The pentagon has five exterior angles which add up to 360°. 360 ÷ 5 = 72. Each exterior angle must equal 72°.
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An exterior angle of a regular polygon can be found by dividing 360° by the number of sides (𝒏). Eg, for an eight-sided shape (octagon), divide 360° by 8 to find the size of an exterior angle. 360 ÷ 8 = 45. Each exterior angle is 45°.
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The exterior angle of a regular 12-sided shape (dodecagon) can be found by dividing 360° by 12. 360 ÷ 12 = 30. Each exterior angle is 30°.

Question

What is the size of the missing exterior angle (\(x\)) of this quadrilateral?

An image of an irregular quadrilateral. Each of the four exterior angles have been drawn and are labelled, one hundred and thirty three degrees, eighty three degrees, ninety six degrees, and x.

Practise measuring interior and exterior angles in polygons

Quiz

Practise measuring interior and exterior angles in polygons with this quiz. You may need a pen and paper to help you with your answers.

Real-life maths

An image of a bird-eye view of the pentagon building. — Image caption,
The Pentagon is shaped like a five-sided polygon.

The Pentagon is a building in The United States of America. It is the headquarters of the Department of Defence.

It was designed this way to reduce the walking time from one office to another.

The building has over 17 miles of corridors, but it takes just seven minutes to walk between any two areas.

If the building had been rectangular, the travel and time taken to get from one area to another would be much greater.

An image of a bird-eye view of the pentagon building. — Image caption,
The Pentagon is shaped like a five-sided polygon.

An image showing two layouts for an office building. Layout one is a rectangle subdivided vertically into five squares. Each is labelled, left to right office A, B, C, D and E. Written below: An arrow from office A to office E, labelled longer distance. Layout two is a pentagon divided into five triangles. The triangles are labelled clockwise from office A , B, C, D to E. An arrow connects office A and E. Written below: shorter distance. — Image caption,
The shape of the Pentagon reduces the time taken to walk between offices.

More on Angles

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