3D problems with Pythagoras and trigonometry

Pythagoras in 3D

• The diagonal of a cuboid with sides a, b and c is √(a² + b² + c²). A 3 × 4 × 12 cuboid has diagonal √(9 + 16 + 144) = √169 = 13.
• The space diagonal of a cube with side s is s√3.
• Find lengths step by step: first the diagonal of the base, then use it with the height.

Angles in 3D

• The angle between a line and a plane is the angle between the line and its projection (shadow) on the plane.
• Cuboid with base 3 × 4 and height 12: the base diagonal is 5, so the angle between the space diagonal and the base satisfies tan θ = 12 ÷ 5, giving θ = 67.4°.
• Pyramids: for a square-based pyramid with base side 6 and height 4 (apex above the centre), the half-diagonal is 3√2, so each sloping edge is √(18 + 16) = √34.
• The angle between two planes is found using lines in each plane that are perpendicular to where the planes meet.

Non-right-angled triangles

• Sine rule: a/sin A = b/sin B = c/sin C. Use it with a pair of opposite sides and angles.
• Cosine rule: a² = b² + c² − 2bc cos A. Use it with two sides and the angle between them, or three sides.
• Area = ½ab sin C. For sides 6 and 8 with an included angle of 30°: ½ × 6 × 8 × sin 30° = 12.

Key terms

Space diagonal

A diagonal passing through the inside of a 3D shape.

Projection

The shadow of a line on a plane.

Plane

A flat surface.

Sine rule

a/sin A = b/sin B = c/sin C.

Cosine rule

a² = b² + c² − 2bc cos A.

Included angle

The angle between two given sides.