Question

What is the number of sides of a regular polygon whose interior angle is of 135 degree each​

Answer 1

The sum of the interior angles of a p-sided polygon is (p-2)×180°.

So each angle of a regular p-sided polygon is (p-2)/p×180°.

Let n be the number of sides of a regular polygon whose interior angles are each 135°.

Then (n-2))n×180° = 135°.

So (n-2)/n×180°/180° = 135°/180°.

So (n-2)/n = 135/180.

So (n-2)/n = ¾.

So (n-2)/n×n = ¾n.

So n-2 = ¾n.

So n-2-¾n+2 = ¾n-¾n+2.

So (1-¾)n = 2.

So (4/4-¾)n = 2.

So (4-3)/4×n = 2.

So ¼n = 2.

So ¼n×4 = 2×4.

So n = 8.

CHECK:

If p = 8, then the magnitude of each side is:

(p-2)/p×180°

= (8-2)/8×180°

= 6/8×180°

= ¾×180°

= 135°. ✓

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