The measure of a regular polygon’s interior angle is four times bigger than the measure of its external angle. How many sides does the polygon have?

Question

geometry problems

Answer 1

The measure of a regular polygon’s interior angle is four times bigger than the measure of its external angle. How many sides does the polygon have?

 

Let θ be the interior angle and α be the exterior angle, then

α + θ = π, and

θ = 4α,   i.e.

θ = 4(π - θ) = 4π - 4θ

θ = 4π/5

Sum of interior angles of an n-sided polygon is: nθ = (n - 2)π, so

n(4π/5 ) = (n - 2)π

n(4/5 ) = (n - 2)

4n = 5n - 10

n = 10

The polygon has 10 sides