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Mathematics

The interior angle of a regular polygon is four times its exterior angle. The number of sides in the polygon is:

  1. 12

  2. 10

  3. 15

  4. 18

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Answer

It is given the interior angle of a regular polygon is four times its exterior angle.

Let exterior angle be a. Then, the interior angle is 4a.

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

⇒ a + 4a = 180°

⇒ 5a = 180°

⇒ a = 180°5\dfrac{180°}{5}

⇒ a = 36°

So, the exterior angle is 36° and the interior angle is:

4a = 4 x 36° = 144°

According to the properties of a polygon, if a polygon has n sides, then each of its interior angles is (2n−4)×90°n\dfrac{(2n - 4) \times 90°}{n}.

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

⇒ (2n - 4) x 90° = 144°n

⇒ 180°n - 360° = 144°n

⇒ 180°n - 144°n = 360°

⇒ 36°n = 360°

⇒ n = 360°36°\dfrac{360°}{36°}

⇒ n = 10

Hence, option 2 is the correct option.

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