The Interior Angles of a Polygon

The Interior Angles of a Polygon

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°

See Also

What is an angle?