Finding the angle between vertices in a three dimensional solid can be tricky. It often helps to construct a triangle between the relevant vertices. Then it is often possible to use trigonometry. Consider the hexagonal pyramid below.

The base is a regular pyramid and all the sides are the same length, 2cm. The pyramid is 5 cm high.

We want to find the anglefor which we need to find the lengthThe base of the pyramid is shown below.

We may cut the base up into six equilateral triangles, so that each angle is 60 and each length is 2 cm The length(O is the centre of the base so FOC is a straight line) is then 4 cm and we may construct the triangle below.

The angleis twice the angleThe triangleis right angled soand

Finding the angleis trickier. We have to find the slant heightand the distance

From the triangle above, Pythagoras Theorem givesW can draw the triangle below to find

We can find the distance AB using the cosine rule

The triangleis then

We can findusing the cosine rule.