3D Trigonometry
Cuboids, diagonals and the angle with a plane
In this note
- 6.4Find the space diagonal of a cuboid or cube with Pythagoras twice.
- 6.4Find the angle between a line and a plane.
- 6.4Work backwards from an angle to a length.
Before you start: Pythagoras and right-angled trigonometry. Next: Pyramids and prisms.
Formula bank
The space diagonal
Every 3D question is a 2D right-angled triangle hiding inside the solid. For the diagonal AG: first find the base diagonal AC (a right-angled triangle on the floor), then use triangle ACG, which stands up with a right angle at C.
General methodHow to find a length in a cuboid
- Find the right-angled triangle that contains the length you want.
- If one of its sides is missing, find it first in another right-angled triangle (usually on the base).
- Pythagoras twice; or in one go \(\sqrt{a^{2}+b^{2}+c^{2}}\).
- Sketch each triangle flat on the page before calculating.
Example A cuboid is 3 cm by 4 cm by 12 cm tall. Find the length of the space diagonal.
Base diagonal first.
- base: \(\sqrt{3^{2}+4^{2}}=5\)
- diagonal: \(\sqrt{5^{2}+12^{2}}=13\)
Answer 13 cm
Try it yourself
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Answer 11.18
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Answer 6.93
The angle between a line and a plane
Shine a light straight down onto the plane: the line casts a 'shadow' (its projection). The angle between the line and the plane is the angle between the line and that shadow. For a cuboid's diagonal AG and the base, the shadow is AC, so the angle is GAC.
General methodHow to find the angle between a line and a plane
- From the top end of the line, drop straight down to the plane. That foot point (C) is where the shadow ends.
- The angle is at the bottom end of the line (A), between the line (AG) and the shadow (AC).
- Triangle ACG has a right angle at C. Find two of its sides.
- SOH CAH TOA, usually tan(height ÷ shadow).
Example For the 3 by 4 by 12 cuboid, find the angle between the space diagonal and the base.
Shadow = base diagonal 5, height 12.
- \(\tan\theta=\dfrac{12}{5}\)
- \(\theta=67.4^\circ\)
Answer 67.4°
Example A cube has side 4 cm. Find the angle between a space diagonal and the base.
Base diagonal \(\sqrt{32}\), height 4.
- \(\tan\theta=\dfrac{4}{\sqrt{32}}\)
- \(\theta=35.3^\circ\)
Answer 35.3°
Example A space diagonal of length 15 cm makes 40° with the base. Find the height of the cuboid.
Height is opposite 40°, the diagonal is the hypotenuse.
- \(h=15\sin40^\circ=9.64\)
Answer 9.64 cm
Try it yourself
Show answer
Answer 26.6
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Answer 47.5
Ways they test it
Type 1 Length inside a cuboid
"Calculate the length AG." (3 marks)
Spot it a cuboid, a line between opposite corners.
- Base diagonal.
- Then the space diagonal.
Type 2 Angle with the base
"Calculate the angle between AG and the base ABCD." (3-4 marks)
Spot it 'angle between [line] and [plane]'.
- Drop from the top to find the shadow.
- tan⁻¹(height ÷ shadow).
For example 3 × 4 × 12
- \(\tan^{-1}(12/5)\)
Answer 67.4°
Type 3 Backwards from the angle
"The angle between PV and the base is 43°. Calculate PT." (3 marks)
Spot it an angle and the diagonal given.
- Right-angled triangle with the diagonal as hypotenuse.
- sin or cos.
Type 4 Angle in a different plane
"Calculate angle HAF." (6 marks)
Spot it a triangle inside the cuboid that is NOT right-angled.
- Find all three sides by Pythagoras.
- Cosine rule.
Mistakes to avoid
- ✗ Using the base EDGE instead of the base DIAGONAL as the shadow✓ the shadow runs corner to opposite corner
- ✗ Angle at the wrong vertex✓ it is where the line meets the plane
- ✗ Rounding the base diagonal✓ keep √ form or full accuracy
Summary
- Space diagonal \(=\sqrt{a^{2}+b^{2}+c^{2}}\).
- Line–plane angle: between the line and its shadow on the plane.
- Always pull out a flat right-angled triangle and solve it in 2D.
If you see …
- 'angle between the line and the base'drop down, find the shadow, tan.
- a triangle with no right anglethree sides, cosine rule.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer 5
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Answer 13
Show answer
Answer 6.40
Medium
Guide me
What is it asking? a length
What do we know? \(\sqrt{16+16+16}\)
Plan \(4\sqrt3\)
Show working
- 6.93
Answer 6.93
Guide me
What is it asking? an angle
What do we know? shadow 5, height 12
Plan \(\tan^{-1}(12/5)\)
Show working
- 67.4
Answer 67.4
Guide me
What is it asking? a length
What do we know? 25 + 16 + 49
Plan \(\sqrt{90}\)
Show working
- 9.49
Answer 9.49
Hard
Guide me
What is it asking? a length
What do we know? diagonal = hypotenuse
Plan \(15\sin40^\circ\)
Show working
- 9.64
Answer 9.64
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\sin40=\frac{h}{15}\) | M1 | ||
| 9.64 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? an angle
What do we know? shadow \(\sqrt{32}\)
Plan \(\tan^{-1}(4/\sqrt{32})\)
Show working
- 35.3
Answer 35.3
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| \(\sqrt{32}\) | M1 | ||
| \(\tan^{-1}\) | M1 | ||
| 35.3 | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? an angle
What do we know? shadow 10
Plan \(\tan^{-1}(5/10)\)
Show working
- 26.6
Answer 26.6
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| AC = 10 | M1 | ||
| 26.6 | A1 | ||
| Total: 2 marks | |||
Can I do this?
- I can find a space diagonal of a cuboid.
- I can find the angle between a line and a plane.
- I can work back from an angle to a length.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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