I'm helping a friend with a 3d design, and can't figure this out. I assumed it would be simple, but For whatever reason, I can't get the angles right on the trapezoids, so that 5 of them will make the outer ring on a pentagon, that would then fit a smaller pentagon on the inside. I want all the trapezoids to be identical - is this just not possible? It should look like a pentagon pie chart, like this(see image). I just can't get the 3d design to match on the angles.
All of the trapezoids are congruent (identical) to each other.
One interior angle of the pentagon is $\frac{180º(5 -2)}{5} = 108º$. Thus the base angles of the trapezoids are $108º/2 = 54º$, and since the trapezoid bases are parallel, the other angles are both $180º - 54º = 126º$.
The $108º/2$ part may need more justification: it looks like it's the case, but how can we be sure? Well, all five triangles with their apex at the centre of the pentagon are congruent because their bases are the same, and all of the adjacent triangles share the other two sides. Thus the triangles share the same base angles, and two of them form the interior angle of the pentagon, so each angle is half of the larger angle.