Pythagoras’ theorem and trigonometry in 3D

Every 3D problem is solved in a flat right-angled triangle hidden inside the solid. Find that triangle, sketch it on its own, and label its sides.

Pythagoras' theorem works in 3D by using it twice. First find a diagonal across a flat face. Then use that diagonal as one short side of a second right-angled triangle.

For a cuboid, the two steps join into one rule: the space diagonal squared is the sum of the squares of the three edges.

Where a solid has a horizontal base and a vertical edge, a line from a base corner to the top of a vertical edge is the hypotenuse of a right-angled triangle standing on the base.

Work in two steps: Pythagoras across the base, then Pythagoras up the vertical edge. Keep the unrounded value from the first step in your calculator and round only the final answer.

When you find a short side, subtract the squares: a² = c² - b².

The angle between a line and a plane is found by dropping a perpendicular from the far end of the line to the plane. For a horizontal plane this is a vertical drop. The angle sits where the line meets the plane, between the line and the plane.

That gives a right-angled triangle: the line is the hypotenuse, the vertical drop is the side opposite the angle, and the side along the plane is adjacent.

In a cuboid, a space diagonal makes its angle with the diagonal of the base, not with an edge. On a calculator, set degree mode and use sin⁻¹, cos⁻¹ or tan⁻¹ on the stored ratio.

In a pyramid on a square base, the vertex P is directly above the centre O of the base. The vertical height PO is at right angles to every line on the base.

The slant height PM runs from the vertex to the midpoint M of a base edge. In triangle POM, the side OM is half the base edge.

To find the angle between PM and the base, use the angle at M in triangle POM, not an angle in a sloping face.

The angle between a sloping edge and the base is a different angle. The edge meets the base at a corner, so the base side of the triangle is half the diagonal of the base, from the centre O to that corner.

Find the whole diagonal first with Pythagoras. Halve it. Then use trigonometry in the triangle formed by the vertical height, the half-diagonal and the sloping edge.