Pearce R. answered 04/02/20
Mathematics & Robotics Mentor
So, there is a trick to these problems. The key to understand is that, because each of these shapes is a bunch of triangles joined together at the middle of the shape, the sum of the angles in the center of the shape is 360 degrees.
So, for the octagon example, angles near the center sum to 360, and there are 8 of them, so each isosceles triangle has one angle that is 360/8, or 45 degrees.
In an isoceles triangle, two angles are the same and the sum of the three angles are 180 degrees. So, 45+2a = 180, a = 67.5 degrees
So, the equation for determining the vertex angles of the isosceles triangles is to divide 360 by the number of triangles (or the number of sides of the large shape)