## Angles in Three Dimensional Solids

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 angle for which we need to find the length The 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 angle is twice the angle The triangle is right angled so and Finding the angle is trickier. We have to find the slant height and the distance From the triangle above, Pythagoras Theorem gives W can draw the triangle below to find  We can find the distance AB using the cosine rule The triangle is then We can find using the cosine rule.  