Question

The ratio of exterior angle to interior angle of a regular polygon is 1:4 .find the number of sides of the polygon

Answer 1

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Step by Step explanation :-
$\frac{exterior \: angles}{interior \: angles} = \frac{1}{4} \\ \\ \frac{360}{n} \div \frac{(n - 2) \times 180} {n} = \frac{1}{4} \\ \\ \frac{360}{n} \times \frac{n}{(n - 2) \times 180} = \frac{1}{4} \\ \\ \frac{360}{180n - 360} = \frac{1}{4} \\ \\ \: 4 \times 360 = 180n - 360 \\ \\ 1440 = 180n - 360 \\ \\ 1440 + 360 = 180n \\ \\ 1800 = 180n \\ \\ n = \frac{1800}{180} \\ \\ n = 10 \: answer$

Hence the polygon is having 10 sides.... ✔✔

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Answer 2

$<b>$
Let the interior angle of the regular polygon be x.
Therefore, the exterior angle is x/4.

Exterior angle + adjacent interior angle = 180°
x/4 + x. = 180°
5x /4. = 180°
x. = 180° * 4/5
= 144°

The interior angle is 144°.
The exterior angle is 36°.

Let n be the number of sides.

n = 360°/ exterior angle
= 360° / 36°
= 10

Ans.= The number of sides of a regular polygon is 10.