The LessonThe interior angles of a polygon are the angles between two sides, inside the shape.
The Sum of the Interior Angles of a PolygonThe 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.
ExampleImagine 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 Regular PolygonRegular 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:
ExampleImagine 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°
The Interior Angles of Different Polygons
|Shape||Sum of Interior Angles||Interior Angle of Regular Polygon|
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°.
Exterior AnglesPolygons 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°