Pizza Slice (XYPIZQ) mini editorial

Consider this diagram. Half of the line segments are drawn in red. The line segments $a_0b_1,b_1a_2,a_2b_3,b_3a_4$ also exist, but are ommited here for clarity.

We do not yet know what angle $x$ is, but we can work out the other angles in terms of $x$. We will be working in radians here. $\pi\ radians = 180\ degrees$

Because the slice is an isosceles triangle, we can calculate:

$c = \frac{\pi-x}{2} = 0.5\pi - 0.5x$

Knowing that all the inner triangles are also isosceles trangles, we can also calculate:

$d = \pi - 2c = x$

$e = c - d = 0.5\pi - 1.5x$

$f = \pi - 2e = 3x$

$g = \pi - c - f = 0.5\pi -2.5x$

$h = \pi - 2g = 5x$

$i = \pi - e - h = 0.5\pi - 3.5x$

$j = \pi - 2i = 7x$

Also because of the isosceles triangle, $x = i$, so we can now derive the value of $x$:

$x = 0.5\pi - 3.5x$

$4.5x = 0.5\pi$

$9x = \pi$

Now we can go back and substitute $\frac{\pi}{9}$ for $x$ and work out all the angles exactly.

We can also see a pattern in the angles. The angles of the tops of the triangles go:

$x,3x,5x,7x..$

while the angles of the sides go

$0.5\pi-0.5x$

$0.5\pi-1.5x$

$0.5\pi-2.5x$

$0.5\pi-3.5x$

$..$

So we can work out the angles of specific triangles with closed form formula, instead of having to calculate them all.