Trigonometry in three dimensions

• Pick out a right-angled triangle inside the solid and redraw it as a flat 2D triangle.
• Often you need one triangle on the base first, then a second, vertical triangle.
• The diagonal of a cuboid: d² = a² + b² + c².
• The angle between a line and a plane is the angle between the line and its projection on the plane.
• Use the method in pyramids, prisms, wedges and problems with vertical masts or towers.

Two steps, or one formula

To find the long diagonal AG of a cuboid, first find the diagonal of the base AC (a right-angled triangle on the base), then use the vertical triangle ACG, which has a right angle at C.

Drop a perpendicular onto the plane

From the top end of the line, drop a perpendicular down to the plane. Join the foot of that perpendicular to the point where the line meets the plane: this is the projection of the line on the plane. The angle you want is between the line and its projection, in a triangle with a right angle at the foot of the perpendicular.

The apex is above the centre

In a right pyramid the apex is vertically above the centre of the base. For a square base of side s, the centre is half a diagonal from each corner (s√2 ÷ 2) and half a side from the middle of each edge.

Wedges, masts and towers

Draw the solid, find the right angles (vertical lines meet horizontal ground at 90°), and redraw each triangle you use flat on its own.