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Trigonometry in 3D and Mixed Problems

Finding lengths and angles in cuboids and pyramids, and combining Pythagoras and the sine and cosine rules.

  • Higher
  • 9 key terms
  • All boards

Learning Objectives

  1. 1Find the length of a diagonal of a cuboid using Pythagoras' theorem in three dimensions.
  2. 2Find the angle between a line and a plane using right-angled trigonometry.
  3. 3Use trigonometry in pyramids, and mix the sine rule, the cosine rule and Pythagoras in one problem.
  4. 4Draw out the right-angled triangle that you need, and label it clearly.

Trigonometry in three dimensions

Three-dimensional problems look hard because the diagram is a flat picture of a solid. The method is to find a right-angled triangle inside the solid, redraw it on its own with its true lengths, and then use Pythagoras' theorem or trigonometry as usual. The angle between a line and a plane is the angle between the line and its shadow on the plane, which is found using a right-angled triangle. Mixed problems also use the sine rule and the cosine rule in non-right-angled triangles.

Diagonals and angles in a cuboid

Break the problem into two right-angled triangles, one flat and one upright.

  • Base diagonal

    For a base of length \(l\) and width \(w\), \(\text{diagonal}^2 = l^2 + w^2\).

  • Space diagonal

    \(d^2 = l^2 + w^2 + h^2\).

  • Angle to the base

    Use the base diagonal as the adjacent side and the height as the opposite side.

  • Always

    Redraw the triangle in two dimensions, and mark the right angle.

A space diagonal

A cuboid has length 12 cm, width 4 cm and height 3 cm. Work out the length of the diagonal from one corner to the opposite corner.

Show the solutionHide the solution
  1. 1 Base diagonal \(AC^2 = 12^2 + 4^2 = 144 + 16 = 160\).
  2. 2 Add the height \(d^2 = 160 + 3^2 = 169\).
  3. 3 Square root \(d = 13\) cm.
  4. 4 Shortcut \(d^2 = l^2 + w^2 + h^2 = 144 + 16 + 9 = 169\).

Answer13 cm

Mixed problems

Many problems combine several ideas, so plan the route before calculating.

  • Right-angled triangle

    Use SOH CAH TOA or Pythagoras.

  • Any triangle

    Use the sine rule with a matching pair, or the cosine rule with two sides and the included angle.

  • Several steps

    Work out a length in one triangle, then use it in the next.

  • Bearings

    A bearing question can often be set up as a triangle, with the angle found from the bearings.

Two triangles

\(ABCD\) is a quadrilateral. \(AB = 4\) cm, \(BC = 4\) cm and angle \(ABC = 90^\circ\). \(CD = 4\) cm and angle \(ACD = 90^\circ\). Work out the length of \(AD\).

Show the solutionHide the solution
  1. 1 First triangle \(AC^2 = 4^2 + 4^2 = 32\), so \(AC = 4\sqrt{2}\) cm.
  2. 2 Second triangle Triangle \(ACD\) has a right angle at \(C\), so \(AD^2 = AC^2 + CD^2\).
  3. 3 Substitute \(AD^2 = 32 + 16 = 48\).
  4. 4 Simplify \(AD = \sqrt{48} = 4\sqrt{3}\) cm.

Answer\(AD = 4\sqrt{3}\) cm

Test yourself

  1. 1

    What is the formula for a space diagonal?

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    \(d^2 = l^2 + w^2 + h^2\).

  2. 2

    How do you find the angle between a line and a plane?

    Show answerHide answer

    Use the right-angled triangle formed by the line, its shadow on the plane, and the vertical.

  3. 3

    What is the foot of the height in a square-based pyramid?

    Show answerHide answer

    The centre of the base.

  4. 4

    What do you do first in a 3D problem?

    Show answerHide answer

    Find and redraw a right-angled triangle.

  5. 5

    Which rule do you use if there is no right angle?

    Show answerHide answer

    The sine rule or the cosine rule.

Exam technique: 3D problems

Planning gets the marks.

  • Redraw

    Draw each triangle by itself, with its true lengths.

  • Mark the right angle

    This shows which side is the hypotenuse.

  • Keep exact values

    Leave surds until the end, so that the answer is exact.

  • Sensible answers

    Check that the lengths are reasonable, and that a space diagonal is longer than any edge.

Summary and exam focus

  • Find a right-angled triangle inside the solid, and redraw it.
  • The space diagonal of a cuboid is \(\sqrt{l^2 + w^2 + h^2}\).
  • The angle between a line and a plane is found using the vertical height and the shadow on the plane.
  • Mixed problems combine Pythagoras, SOH CAH TOA, and the sine and cosine rules.

Exam focus

A cuboid has a base 4 cm by 3 cm and a height of 5 cm. Work out the angle between the diagonal \(AG\) and the base. (4 marks) (4 marks)

The base diagonal is \(\sqrt{4^2 + 3^2} = 5\) cm. In the right-angled triangle, \(\tan\theta = \dfrac{5}{5} = 1\), so \(\theta = 45^\circ\).

Key terms

The words this lesson expects you to use. Each one is linked from the first place it appears above.

Space diagonal
A line from one corner of a cuboid to the opposite corner.
Plane
A flat surface.
Apex
The top point of a pyramid.
Cuboid
A solid with six rectangular faces.
Pyramid
A solid with a flat base and sloping triangular faces meeting at the apex.
Hypotenuse
The longest side of a right-angled triangle.
Exact value
A value written with surds, not as a decimal.
Angle between a line and a plane
The angle between the line and its shadow on the plane.
Perpendicular
At \(90^\circ\).

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Flashcards 16 cards to flip through Exam questions 6 questions, 24 marks with answers
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