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Angles in a Polygon

Angles in a Polygon

A polygon is a shape with straight edges. A shape will either be a regular polygon or an irregular polygon. Regular Polygons have edges that are all of the same length, and angles that all have the same value. Irregular Polygons have at least one angle or length that is different to the others.

Polygons have more than two sides: triangles, rectangles and octagons are all polygons.

The inside angle on a polygon is known as an interior angle. An exterior angle is formed by extending one of the sides, then determining the amount of turn needed to form the new side.

Interior and Exterior angles

An Interior Angle and an Exterior Angle together form a straight line, and therefore add to 180º.

The sum of the exterior angles of a polygon always add up to 360º.

Sum of Exterior angles

The sum of the interior angles can be worked out by first dividing the polygon into triangles:

Triangles forming interior angles

In the example, there are 5 triangles that can be drawn in the 7-sided shape. The sum of the angles is 5 x 180 = 900º.

The sum of the interior angles = (number of sides - 2) x 180

Example 1

What is the value of the missing angle A in this irregular polygon?

Missing exterior angle in a pentagon

The sum of the exterior angles is 360º

360 - 42 - 82 - 78 - 76 = 82º

Answer: 82º

Example 2

An octagon has six interior angles of 147º each.

The seventh interior angle is 104º.

What is the size of the eighth angle?

Interior Angles = (sides - 2) x 180
= (8 - 2) x 180
= 1080
Eighth angle = 1080 - 6 x 147 - 104
= 100º

Answer: 100º