Question

The ratio of an interior angle to an exterior angle of a regular polygon is 5 : 2. What is the number of sides of the polygon ?​

Answer 1

Answer:

$\huge\underline\bold {Answer:}$

Interior angle of polygon of n sides = (n – 2) × 180°/n

Exterior angle of a polygon of n sides = 360°/n

Given, (n – 2) × 180°/n : 360°/n = 5 : 2

=> (n – 2) × 180°/n × 360°/n = 5/2

=> (n – 2)/2 = 5/2

=> 2(n – 2) = 10

=> 2n – 4 = 10

=> 2n = 14

=> n = 7.

Therefore number of sides of the polygon is 7.

Answer 2

Answer:

the exterior angle of a regular polygon is equal to 360 / n, where nis the number of sides.

the interior angle of a regular polygon is equal to 180 - 360 / n, where n is the number of sides.

the ratio of the interior angle of the polygon to the exterior angle of the polygon is equual to 5/2.

thie means that (180 - 360 / n) / (360 / n) is equal to 5 / 2.

take the cross product to get:

5 * (360 / n) is equal to 2 * (180 - 360 / n)

solve for n to get n = 7.

when n = 7, the exereior angle is 360 / 7 = 51.42857143 degrees.

 

when n = 7, the interior angle is 180 - 51.42857143 degrees which is equal to 128.5714286 degrees.

the ratio of the interior angle to the exterior angle is equal to 128.5714286 / 51.42857143 which is equal to 2.5.

Multiply the numerator and denominator of 2.5 / 1 to get 5 / 2.

the solution is confirmed as good.

another calculation for the interioa angle would be (n-2) * 180 / n.

when n = 7, that becomes 5 * 180 / 7 which becomes 128.5714286.

the interior angle can be calculated in both ways.

it's the same either way.

Step-by-step explanation:

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