3 Dimensional Problems

MathsDirect

The Angle between a line and a plane

How do you find the angle between the red line AF and the base of the cuboid.

When finding the angle between a line and a plane, what you want is the angle between the line and the line directly below it in the plane.

In this case, the line AH is directly below AF. This line is called the projection of AF onto the plane AEHD

To understand what we mean by directly below, consider the diagram,

The plane through the two lines, represented by the triangle, is at right angles to the planeAEHD.

You can use simple trigonometry to find the angle between the two lines. For example

The point P is directly above C.

AB = 5cm ; AD = 8cm ; CP = 3cm

Find the angle between AP and the base of the pyramid.

We need to find the angle PAC

All we need do is use Pythagoras' to find the lengths AC and AP and then us cosine.

First find the length AC

You can use the length that you have calculated for AC to find the length AP.

(Actually, you do not need to calculate AC. You are better off keeping the answer in Surds).

Now simply put the two lengths into the formula for cos, to find the angle between the lines

Finally, remember to answer the question. You were not asked for the angle between AC and AP. Clearly state

The angle between the line AP and the base of the pyramid is 17.64^o.

On the next page, we will look at finding the angle between two planes.