Question

Find number of sides of a polygon exterior angles and interior angles ratio 1 : 5

Answer 1

Sol:
Let S1 be the number of sides of 1st polygon.

And S 2 be the number of sides of 2nd polygon.

Given S1/ S2 =  1 / 2 --------------(1)
But

measure of interior angle of 1st  polygon / measure of interior angle of 2nd polygon = 3 / 4 -----(2)

We know that in a regular polygon

Measure of each interior angle = Sum of the interior angles / No.of sides

                                            = ( n-2) x 180 / n  where n is No.of sides
 
Thus, [( S1 - 2) x 180 / S1  ] / [( S2 - 2) x 180 / S2  ] = 3 / 4

[( S1 - 2) S2 ] / [( S2 - 2) S1  ] = 3 /4

[( S1 - 2) 2] / [( S2 - 2) 1] = 3 /4

[( S1 - 2) ] / [( S2 - 2) ] = 3 / 8

8 ( S1 - 2)  = 3 ( S2 - 2)  But S2 = 2 S1

8 S1 – 6 S1  = - 6 + 16

2 S1  = 10

 S1  = 5 sides

Thus S2 = 2 S1

S2 = 2 x 5 = 10 sides

There fore  number of sides of each polygon is 5 and 10.