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The Interior Angles of a Polygon

(KS2, Year 6)

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: polygons_sum_of_interior_angles

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. interior_angles_of_a_triangle_example

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°. 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: regular_polygons_interior_angles

Example

Imagine you wanted to find an interior angle of a regular quadrilateral.

regular polygons interior angles example

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°. Learn more about finding 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 2-dimensional shape with straight sides.

Why Does the Formula Work?

The interior angles of a three-sided shape (a triangle) add up to 180°.
To make a four-sided 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 five-sided 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°.

Every time another side is added to a shape, another triangle is added, and another 180° is added to the sum of the interior angles. Note that the number of triangles is always 2 less than the number of sides. That is, there are n − 2 triangles for every n triangles. Since there are 180° for every triangle, the sum of the interior angles is (n − 2) × 180° for every n sides.