The ratio between the exterior angle and the interior angle of a regualr polygon is 1 : 4. Find the number of sides in the polygon.

It is given that the ratio between the exterior angle and the interior angle of a regular polygon is 1 : 4.

Let the common factor be a.

Then:

Interior angle = 4 x a = 4a

Exterior angle = 1 x a = a

We know that sum of the interior angle and the exterior angle is 180°.

⇒ 4a + a = 180°

⇒ 5a = 180°

⇒ a = $\dfrac{180°}{5}$

⇒ a = 36°

Thus:

Interior angle = 4a = 4 x 36° = 144°

Exterior angle = a = 36°

According to the properties of polygons, each interior angle of a regular polygon with n sides is $\dfrac{(2n - 4) \times 90°}{n}$

⇒ $\dfrac{(2n - 4) \times 90°}{n}$ = 144°

By cross-multiplying, we get

⇒ (2n - 4) x 90° = 144°n

⇒ 180°n - 360° = 144°n

⇒ 180°n - 144°n = 360°

⇒ 36°n = 360°

⇒ n = $\dfrac{360°}{36°}$

⇒ n = 10

Hence, the number of sides is 10.