PrepWithTee owl logo PrepWithTee
3D Trigonometry

3D Trigonometry

Pyramids and prisms in 3D

Pyramids Extended 7 min read

On this page
  1. In this note
  2. Formula bank
  3. Pyramids
  4. Wedges and prisms
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 6.4: Pythagoras and trigonometry in 3D (pyramids and prisms)
  • 6.4Find edges and heights of a pyramid using the centre of its base.
  • 6.4Find the angle between a pyramid's edge (or a line on its face) and the base.
  • 6.4Find lengths and angles in a wedge or other prism.

Before you start: Cuboids, diagonals and the angle with a plane.

Formula bank

Centre of a square baseLearn it\(AM=\tfrac12\times\text{diagonal}=\tfrac12\sqrt{a^{2}+b^{2}}\)M = where the diagonals cross; the apex is above M.Use it when a right pyramid.
Slant edgeLearn it\(VA=\sqrt{AM^{2}+VM^{2}}\)VM = vertical height.Use it when edge length.
Edge to baseLearn it\(\tan\theta=\dfrac{VM}{AM}\)Use it when angle between VA and the base.

Pyramids

Square-based pyramid: the angle between edge VA and the base is VAMθABCDVM
Apex V is directly above M, the centre of the base.

In a right pyramid the apex V is directly above the centre M of the base. Triangle VAM stands up with a right angle at M: its base AM is HALF a diagonal of the base, its height is VM.

General methodHow to answer any pyramid question

  1. Find the base diagonal by Pythagoras, then HALVE it: that is AM.
  2. Triangle VAM is right-angled at M. Two of VA, AM, VM give the third (Pythagoras).
  3. Angle between edge VA and the base: angle VAM, use tan (VM ÷ AM).
  4. Line from V to the MIDDLE of a base edge? Its shadow is half the side length, not half the diagonal.

Example A pyramid has a square base of side 6 cm and vertical height 9 cm. Find the length of a slant edge and the angle it makes with the base.

\(AM=\tfrac12\sqrt{72}=\sqrt{18}\).

  1. \(VA=\sqrt{18+81}=9.95\)
  2. \(\tan\theta=\dfrac{9}{\sqrt{18}}\Rightarrow\theta=64.8^\circ\)

Answer 9.95 cm and 64.8°

Example Same pyramid: find the angle between the base and the line from V to the midpoint of a base edge.

The shadow now runs from the edge midpoint to M: half of 6 = 3.

  1. \(\tan\theta=\dfrac{9}{3}\)
  2. \(\theta=71.6^\circ\)

Answer 71.6°

Try it yourself

1
A pyramid has a rectangular base 10 cm by 24 cm and height 9 cm. Length AM to the centre of the base (cm)?
Show answer

Answer 13

2
Same pyramid: angle between a slant edge and the base (1 d.p.)?
Show answer

Answer 34.7

Wedges and prisms

Wedge: AF and its shadow AC on the baseθABCDEF
AF slopes up across the wedge; its shadow AC lies on the base.

A wedge (triangular prism) has a sloping face. The line AF from a bottom corner to the far top corner works exactly like a cuboid's space diagonal: drop from F to C, then use triangle ACF.

Example A wedge has a rectangular base ABCD with AB = 10 cm and BC = 6 cm. The vertical edge CF = 4 cm. Find AF and the angle between AF and the base.

Base diagonal AC first.

  1. \(AC=\sqrt{100+36}=\sqrt{136}=11.66\)
  2. \(AF=\sqrt{136+16}=12.33\)
  3. \(\tan\theta=\dfrac{4}{\sqrt{136}}\Rightarrow\theta=18.9^\circ\)

Answer 12.33 cm and 18.9°

Try it yourself

1
Same wedge: angle between the sloping edge BF and the base (1 d.p.)?
Hint
BF's shadow is BC = 6.
Show answer

Answer 33.7

Ways they test it

Type 1 Pyramid edge or height

"Calculate the height of the pyramid." (4 marks)

Spot it slant edge given, height wanted (or the reverse).

  1. Half-diagonal AM.
  2. Pythagoras in VAM.

Type 2 Pyramid angle

"Calculate the angle between VA and the base." (4 marks)

Spot it edge and base.

  1. AM and VM.
  2. \(\tan^{-1}(VM/AM)\).

For example Base 6, height 9

  1. \(AM=\sqrt{18}\)

Answer 64.8°

Type 3 Wedge angle

"Calculate the angle between AF and the base ABCD." (3 marks)

Spot it a prism with a sloping face.

  1. Shadow on the base.
  2. tan.

Type 4 Triangular base

"The base is an equilateral triangle; O is its centre…" (several marks)

Spot it a tetrahedron-style pyramid.

  1. They give or lead you to the distance from O to a corner.
  2. Then the same right-angled triangle.

Mistakes to avoid

  • ✗ Using the full diagonal for AM✓ HALF the diagonalM is the centre.
  • ✗ Using half the diagonal for a line to an edge MIDPOINT✓ half the side length
  • ✗ Treating the slant edge as the height✓ the height is vertical, from V to M

Summary

  • Right pyramid: apex above the centre M; AM = half the base diagonal.
  • Triangle VAM (right angle at M) gives edges, heights and the angle with the base.
  • Wedge lines work like cuboid diagonals: drop to the base, find the shadow, tan.

If you see …

  • 'pyramid', 'vertically above the centre'half-diagonal, then triangle VAM.
  • 'wedge' or 'prism' with a sloping lineshadow on the base.

Practice

Practice: easy, medium, hard

Easy

1
A square base has side 6 cm. Half the diagonal (2 d.p.)?
Show answer

Answer 4.24

2
Rectangle 10 by 24. Diagonal (cm)?
Show answer

Answer 26

3
Wedge base 10 by 6. Base diagonal (2 d.p.)?
Show answer

Answer 11.66

Medium

1
Square base 6 cm, height 9 cm. Slant edge (2 d.p.)?
Guide me

What is it asking? a length

What do we know? \(AM^2=18\)

Plan \(\sqrt{18+81}\)

Show working
  1. 9.95

Answer 9.95

2
Base 10 by 24, height 9. Slant edge (2 d.p.)?
Guide me

What is it asking? a length

What do we know? AM = 13

Plan \(\sqrt{169+81}\)

Show working
  1. 15.81

Answer 15.81

3
Wedge: AB 10, BC 6, CF 4. Length AF (2 d.p.)?
Guide me

What is it asking? a length

What do we know? AC² = 136

Plan \(\sqrt{136+16}\)

Show working
  1. 12.33

Answer 12.33

Hard

1
Square base 6 cm, height 9 cm. Angle between a slant edge and the base (1 d.p.)?
Guide me

What is it asking? an angle

What do we know? AM \(=\sqrt{18}\)

Plan \(\tan^{-1}(9/\sqrt{18})\)

Show working
  1. 64.8

Answer 64.8

PartAnswer / workingMarksGuidance
\(AM=\sqrt{18}\)M1
\(\tan^{-1}\)M1
64.8A1
Total: 3 marks
2
Same pyramid: angle between the base and the line from V to the midpoint of a base edge (1 d.p.)?
Guide me

What is it asking? an angle

What do we know? shadow = 3

Plan \(\tan^{-1}3\)

Show working
  1. 71.6

Answer 71.6

PartAnswer / workingMarksGuidance
3 usedM1
71.6A1
Total: 2 marks
3
Wedge: AB 10, BC 6, CF 4. Angle between AF and the base (1 d.p.)?
Guide me

What is it asking? an angle

What do we know? shadow \(\sqrt{136}\)

Plan \(\tan^{-1}(4/\sqrt{136})\)

Show working
  1. 18.9

Answer 18.9

PartAnswer / workingMarksGuidance
\(AC=\sqrt{136}\)M1
18.9A1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

The multiple-choice and guided questions need JavaScript. The concept checks and exam practice below work without it.

Your progress on this note is saved when JavaScript is on.