What is the measure of an interior angle of a regular 9-gon?

Answer 1

Imagine connecting all the vertices of the regular 9-gon to the centre of the 9-gon. We have now created 9 triangles, so the sum of all their interior angles is 9*180 degrees = 1620 degrees. Notice that the sum of these angles is exactly the sum of the interior angles of the 9-gon, except we have created extra angles around the centre point. These extra make a circle, so they must add up to 360 degrees. Therefore is we substract 360 degrees from 1620 degrees we get the sum of all the interior angles of a regular 9-gon: 1620 degrees - 360 degrees = 1260 degrees. So finally, since there are 9 vertices, every single vertex must have an interior angle of 1260 degrees / 9 = 140 degrees.