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The Interior Angles of a Polygon
(KS2, Year 6)
The Lesson
The interior angles of a polygon are the angles between two sides, inside the shape.The Sum of the Interior Angles of a Polygon
The sum of the interior angles of a polygon is given by the formula:In this formula, n is the number of sides of the polygon. The formula tells you what the interior angles of a polygon add up to.
Example
Imagine you wanted to find the sum of the interior angles of a triangle.A triangle has 3 sides, so n = 3. Using the formula:
Sum of interior angles = (n − 2) × 180°
Sum of interior angles = (3 − 2) × 180°
Sum of interior angles = 1 × 180°
Sum of interior angles = 180°
The interior angles of a triangle add up to 180°.
Read more about how to find the sum of the interior angles of a polygon
The Interior Angles of a Regular Polygon
Regular polygons have equal interior angles. There are as many interior angles as there are sides. To find each interior angle in a regular polygon, divide the sum of the interior angles by the number of sides. The formula for each of the interior angles of a regular polygon is:Example
Imagine you wanted to find an interior angle of a regular quadrilateral.A square is a regular quadrilateral. It has 4 sides, so n = 4. Using the formula:
Interior angle = (n − 2) × 180° ÷ n
Interior angle = (4 − 2) × 180° ÷ 4
Interior angle = 2 × 180° ÷ 4
Interior angle = 360° ÷ 4
Interior angle = 90°
Each angle in a square is 90°.
Read more about how to find the interior angle of a regular polygon
The Interior Angles of Different Polygons
Shape  Sum of Interior Angles  Interior Angle of Regular Polygon 

Triangle  180°  60° 
Quadrilateral  360°  90° 
Pentagon  540°  108° 
Hexagon  720°  120° 
Heptagon  900°  128.57° 
Octagon  1080°  135° 
Nonagon  1260°  140° 
Decagon  1440°  144° 
Dodecagon  1800°  150° 
What Is a Polygon?
A polygon is a 2dimensional shape with straight sides.Why Does the Formula Work?

The interior angles of a threesided shape (a triangle) add up to 180°.

To make a foursided shape (a quadrilateral), replace a side of the triangle with two sides.
This shape is made up two triangles, and so the sum of the interior angles is 2 × 180° = 360°.

A fivesided shape (a pentagon) can be made by adding a third triangle, adding another 180° to the sum of the interior angles, so the sum of the interior angles is 3 × 180° = 540°.
Exterior Angles
Polygons have exterior angles as well as interior angles.The sum of the exterior angles of a polygon is 360°. Read more about the exterior angles of a polygon
An Interior and Exterior Angle in a Polygon Add Up to 180°
An interior and exterior angle in a polygon add up to 180°.Read more about how the interior and exterior angle of a polygon add up to 180°
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