3D problems: angles with planes

In 3D the trick is to find a right-angled triangle that contains the angle you want, then draw it flat.

Practise this lesson free (6 questions)

A line and a plane

The angle between a line and a plane is the angle between the line and its projection (its shadow straight down) on the plane.
Cuboid 8 cm by 6 cm by 5 cm high. Base diagonal = √(82 + 62) = 10 cm.
The space diagonal makes angle θ with the base: tan θ = 510, θ = 26.6°.
Space diagonal = √(82 + 62 + 52) = 11.18 cm.

Two planes

The angle between two planes is measured between two lines, one in each plane, both perpendicular to the common edge at the same point.
A square-based pyramid has base 10 cm and vertical height 6 cm. For a sloping face and the base, use the line from the centre of the base to the midpoint of an edge (5 cm):
tan θ = 65, θ = 50.2°.

A slant edge and the base

For a slant edge, the projection runs from a corner to the centre: half the diagonal = 12 × 10√2 = 7.071 cm.
tan θ = 67.071, θ = 40.3°.
The face angle is bigger than the edge angle, because the face rises over a shorter distance.
Worked example

A cuboid is 8 cm by 6 cm by 5 cm high. Find the angle between a space diagonal and the base.

  1. Base diagonal = √(64 + 36) = 10
  2. tan θ = 510
  3. θ = 26.6°

Answer: 26.6°

Key idea

Line and plane: use the line's projection on the plane. Two planes: use lines in each plane perpendicular to the common edge. Draw the right-angled triangle flat and use trigonometry.

Key words
Perpendicular
At right angles. Perpendicular lines have gradients whose product is −1.
tan θ
tan θ = sin θ/cos θ

All International GCSE Further Pure Mathematics key terms and definitions →

Quick check

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Extra practice

2 more questions

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  1. A cuboid is 12 cm by 5 cm by 4 cm high. Find the length of the diagonal of the base.

    Show the answer

    Answer: 13 cm

    √(122 + 52) = √169 = 13 cm.

  2. Plane ABCD is horizontal and plane ABEF slopes up from the edge AB. Which lines do you use to measure the angle between the planes?

    1. The normals to AB at A and B
    2. Two lines meeting at the same point on AB, one in each plane, both perpendicular to AB
    3. Any line in each plane
    4. The edge AB and any line in ABEF
    Show the answer

    Answer: Two lines meeting at the same point on AB, one in each plane, both perpendicular to AB

    The angle between planes uses lines perpendicular to the common edge, one in each plane.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.

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