How many sides does a regular polygon have whose interior and exterior angles are in the ratio 3:1?

Answer 1

interior to exterior is 3 to 1

3x+x=4x

4x=180 because an interior angle and its corresponding exterior angle sum to 180°

therefore x=180/4

x=45°

the interior angles all measure 3*45=135°

to find the number of degrees in an n-gon, use the formula (n-2)*180

so, (n-2)*180=n*135

180n-360=135n

180n-135n=360

45n=360

n=360/45

n=8 sides, an octagon

a simpler solution would be to use the following fact: the sum of the exterior angles of any polygon is always 360°

therefore because each exterior angle measures 45°, 360/45=8 sides again