hi Psalmycole

When you want the angle between two planes, you have to imagine you are looking along the line of intersection (VB). The lengths that you see are the lengths of lines in each plane that are at 90 degrees to VB.

So I looked at the triangle VBC. We know the length of BC. I had to calculate VB ( = VC ) from the triangle VFC.

As VBC is isosceles I used cosine in the right angled triangle made by splitting VBC in half.

Then I used BGC = 90 and sines to calculate GC ( = GA as the faces are the same).

The formula for VC is not simple so neither is cos(VBC). Then switching to sine makes the formula very complicated.

I followed it through and I do not get

My expression is much more complicated.

Using your formula and with a = 4 and h = 3.265986 (which will make a regular pyramid = tetrahedron ) I got cos AGC = -5.9375 which is not possible!

I used my formulas and got angle = 70.528779 which is the Wiki value.

How did you get your expression?

I'll check your working.

Bob