Question

If sum of all the interior angles of a regular polygon is `10` times sum of all exterior angles, find the measure of interior angles?

Answer 1

Each interior angle is `163.bar63^@` or `163.6363......^@`

Explanation:

In a regular polygon, all interior angles are equal and

all exterior angles too are equal.

As sum of interior angle of a polygon and exterior angle is always `180^@`,

if polygon has `n` sides, let each exterior angle be `E^@` and interior angle be `I^@`. Then sum of all interior angles is `nI` and sum of all extrerior angles is `nE` and

`nI=10xxnE` i.e. `I=10E`

but `I+E=180` i.e. `10E+E=180` and `E=(180/11)^@=16.3636...^@=16.bar36`

and each interior angle is `180-16.bar36=163.bar63`

here the bar above a set of numbers indicates that the set of numbers repeat endlessly.

Additional information: As sum of exterior angles is always `360^@`, we have `180/11xxn=360` or `n=22` i.e. regular polygon has `22` sides.