Question

The exterior angle of a regular polygon is equal to one third of the interior angle .calculate the number of sides of the polygon

Answer 1

Facts:

The sum of exterior angles of a polygon is 360°

The formula for calculating the size of an exterior angle is: exterior angle of a polygon = 360°/ number of sides.

Interior Angle = 180° − Exterior Angle

In this case:

E = exterior angle

I = interior angle

n = number of sides

The exterior angle of a regular polygon is equal to one third of the interior angle means:

E = I / 3

Multiply both sides by 3

3 E = I

I = 3 E

Interior Angle = 180° − Exterior Angle means:

I = 180° − E

Since I = 3 E

3 E = 180° − E

Add E to both sides

4 E = 180°

Divide both sides by 4

E = 180° / 4

E = 45°

I = 180° − E

I = 180° − 45°

I = 135°

The formula for calculating the size of an exterior angle is: exterior angle of a polygon = 360°/ number of sides means:

E = 360°/ n

Since E = 45°

45° = 360°/ n

Multiply both sides by n

45° • n = 360°

Divide both sided by 45°

n = 360°/ 45°

n = 8

Check of results:

E = 360°/ n

E = 360°/ 8

E = 45°

I = 180° − E

I = 180° − 45°

I = 135°

E = 45° = 135° / 3

E = I / 3

Answer 2

360/n = 1/3 * 180(n-2)/n
n = 8