Exterior Angles of a Polygon

Exterior Angles of a Polygon

How to solve the exterior angles of a polygon problems: formula, proof, examples, and their solutions.

Formula

The sum of the measures of the exterior angles of an n-gon is 360.

The sum of the measures of the exterior angles of an n-gon is 360.

Proof

Deriving the formula from the interior angles of a polygon formula and the exterior angles of a polygon formula.

For an n-gon,
there are n of interior angles (purple)
and n of exterior angles (blue).

And there are n of linear pairs. (purple & blue)

Sum of interior angles: 180(n - 2)
Sum of exterior angles: (sum) ← wanted
Sum of the linear pairs: 180n

So 180(n - 2) + (sum) = 180n.

Interior angles of a polygon

Example 1

Find the value of x. Exterior angles of a quadrilateral: 3x + 40, 2x + 10, 90, and 120.

The given angles are the exterior angles of a quadrilateral.

So (3x + 40) + (2x + 10) + (120) + (90) = 360.

Example 2

Find the measure of an exterior angle of a regular pentagon.

A regular pentagon has 5 congruent sides
and 5 congruent angles.

So the measure of an exterior angle
of a regular pentagon is 360/5 = 72.

Example 3

The measure of an exterior angle of a regular polygon is 60. Find the number of sides of the polygon.

The regular polygon has n congruent sides
and n congruent angles.

And the measure of an interior angle
of the regular polygon is 60.

So 360/n = 60.