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The Interior Angles of a Polygon (Mathematics Lesson)

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:



where n is the number of sides of the polygon.

The formula tells you what the interior angles of a polygon add up to.

How to Find the Sum of the Interior Angles of a Polygon

Question: What is the sum of the interior angles of a triangle?



A triangle has 3 sides, so n = 3.

Using the formula.



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:

How to Find the Interior Angle of a Regular Polygon

Question: What is the interior angle of a regular quadrilateral?



A square is a regular quadrilateral. It has 4 sides, so n = 4.

Using the formula.



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°
Octogon 1080° 135°
Nonagon 1260° 140°
Decagon 1440° 144°
Dodecagon 1800° 150°

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Note
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 with two sides.

This shape is made up two triangles, and so the interior angles are 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.



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.

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°