The ratio of the measure of an interior angle to an exterior angle of a regular polygon is… — JAMB UTME

The ratio of the measure of an interior angle to an exterior angle of a regular polygon is $5 : 1$ . How many sides does the polygon have?

The polygon has 12 sides.

The interior angle and exterior angle of a regular polygon are supplementary, adding up to

180^\circ

. For a ratio of

5:1

, the exterior angle is

\frac{1}{5+1} \times 180^\circ = 30^\circ

. Using the exterior angle formula

n = \frac{360^\circ}{\text{exterior angle}}

, the number of sides is

\frac{360^\circ}{30^\circ} = 12

Set up an equation using the linear pair relationship between interior and exterior angles.

5x + 1x = 180^\circ \implies 6x = 180^\circ \implies x = 30^\circ

At any vertex of a regular polygon, the interior angle and exterior angle lie on a straight line and are supplementary (

180^\circ

Identify the measure of the exterior angle.

Exterior angle =

30^\circ

The exterior angle corresponds to

1

part of the

5:1

ratio, which equals

x = 30^\circ

Calculate the number of sides

n

of the polygon.

n = \frac{360^\circ}{30^\circ} = 12

The sum of the exterior angles of any convex polygon is

360^\circ

, so

n = \frac{360^\circ}{\text{exterior angle}}

Interior and exterior angles of regular polygons

Question