How many regular polygon have both interior and exterior angles that are integers?

Answer 1

To work out the sum of all the interior angles of a shape, we use the formula:

180(n-2)

In this formula, n = the number of sides the polygon has.

This formula only works out the sum of the interior angles, so to finish it off, we divide the answer by n. The number you get from that is equal to one interior angle of the particular polygon you put in the formula.

To work out the exterior angles, you need to know that:

Exterior angle + interior angle = 180 degrees.

So to get the exterior angle, all we have to do is take the interior angle from 180 degrees.

Triangle:

180(3-2) = 180 degrees

180/3 = 60 degrees

180 - 60 = 120 degrees

Interior angle: 60 degrees

Exterior angle: 120 degrees

Square:

180(4-2) = 360 degrees

360/4 = 90 degrees

180 - 90 = 90 degrees

Interior angle: 90 degrees

Exterior angle: 90 degrees

Pentagon:

180(5-2) = 540 degrees

540/5 = 108 degrees

180 - 108 = 72 degrees

Interior angle: 108 degrees

Exterior angle: 72 degrees

You could do this for a long time and check every single polygon up to a 360 sided shape, but that would take ages. If you think about it logically, we already know that:

Triangle: Int angle = 60 degrees Ext angle = 120 degrees Integers? YES

Square: Int angle = 90 degrees Ext angle = 90 degrees Integers? YES

Pentagon: Int angle = 108 degrees Ext angle = 72 degrees Integers? YES

Do that for as many as you can, and you'll get a decent idea of which ones do and which ones don't. Just a hint: The more sides a polygon has, the less chance that there'll be an integer for it's interior and exterior angles.