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Maths

Interior and exterior angles

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Interior and exterior angles

​​In a nutshell

Interior and exterior angles refer to specific angles in a shape (or polygon). There are formulae that can be used to calculate the interior and exterior angles of regular polygons.


Definitions

A polygon is another word to describe a 2D shape.

A regular polygon is a 2D shape that has equal sides and angles.

An interior angle of a polygon is the inner angle formed by two sides of a shape.

An exterior angle of a polygon is the formed by extending one side of the shape past the vertex.



Example 1


Maths; Lines and angles; KS3 Year 7; Interior and exterior angles


In the diagram above, the angles α\alpha and 75°75\degree are internal angles. The angle 135135^\circ is an external angle.


Note: It is worth noticing that an interior and exterior angle are two angles on a straight line, which means that they add together to make 180°180\degree.



Finding the interior angles of a regular polygon

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

θ=(n2)×180\theta=(n-2)\times180​​


Where θ\theta​ is the sum of the interior angles, and nn​ is the number of sides.

The interior angles of a regular polygon are equal. Therefore, to find the size of one interior angle, divide the value of θ\theta by the number of sides.


Example 2

What is the size of an interior angle of a regular nonagon?


A nonagon has 99 sides, so n=9n=9. The sum of the interior angles is therefore given to be:

θ=(n2)×180=(92)×180=7×180\theta=(n-2)\times180=(9-2)\times180=7\times180​​

θ=1260°\theta=1260\degree​​


To find the size of one interior angle, divide this number by 99:

1260÷9=1401260\div9=140​​


The size of one interior angle of a nonagon is 140°\underline{ 140\degree}.


Finding the exterior angles of a regular polygon

Exterior angles of a regular polygon always add up to 360°360\degree. Therefore, the formula for one exterior angle is given to be:

θ=360n\theta=\dfrac{360}{n}​​


Where θ\theta​ is the size of one exterior angle, and nn​ is the number of sides.


Example 3

What is the size of one exterior angle of a regular nonagon?


Substitute n=9n=9 into the formula to give:

θ=3609=40\theta=\dfrac{360}{9}=40​​


The size of one exterior angle of a regular nonagon is 40°\underline{ 40\degree}.


Note: Notice how the interior and exterior angle of a regular nonagon adds up to 180°180\degree.


Want to find out more? Check out these other lessons!

Angle rules

Frequently Asked Questions (FAQ)

FAQs

  • Question: What is the formula for an exterior angle of a regular polygon?

    Answer: The exterior angle of a regular polygon is found by calculating 360/n, where n is the number of sides.

  • Question: What is the formula for the sum of interior angles?

    Answer: The sum of interior angles of a polygon is (n-2)x180, where n is the number of sides.

  • Question: What are interior and exterior angles?

    Answer: An interior angle of a polygon is the inner angle formed by two sides of the shape. An exterior angle of a polygon is the angle formed by extending one of the lines past the vertex.

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