3d Pythagoras And Trigonometry

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Lesson notes

3D Pythagoras Theorem

3D Pythagoras theorem: d² = x² + y² + z², where d is the distance between two points and x, y, z are distances in three perpendicular directions.
You can always break a 3D problem into two 2D right-angled triangles instead of using the 3D formula.
The 3D formula is not given in the exam; you must derive it or use 2D steps.
Example: In a cuboid of dimensions 3 cm × 4 cm × 6 cm, the longest diagonal is √(3²+4²+6²) = √61 ≈ 7.81 cm.

Cone: base radius and height

Using Pythagoras in 3D Shapes

Identify right-angled triangles within the 3D shape (e.g., on faces or cross-sections).
Redraw the 2D triangle flat on the page to apply Pythagoras or trigonometry.
Find missing lengths step by step: e.g., first find a face diagonal, then the space diagonal.
Common shapes: cuboids, prisms, pyramids with square bases.

Distance between two vertices of a cuboid

Trigonometry (SOHCAHTOA) in 3D

Use SOHCAHTOA in right-angled triangles formed within the 3D shape.
Look for triangles that include the required angle or side; you may need to combine multiple triangles.
Always draw the relevant 2D triangle flat to avoid confusion.
Example: In a cuboid, to find the angle between a diagonal and a face, form a triangle with the height perpendicular to that face.

Cone: angle at the base

Angle Between a Line and a Plane

The angle between a line and a plane is the angle between the line and its projection onto the plane.
Construct a right-angled triangle where the height is perpendicular to the plane.
Use SOHCAHTOA to find the angle; the perpendicular side is often a vertical edge.
Tip: Imagine lowering a vertical fishing line from the line to the plane to visualise the right angle.

Angle between a line and a plane in a cuboid

Angle Between Two Lines or Planes

The angle between two lines in 3D can be found by placing them in a common triangle.
The angle between two planes is found by considering the angle between lines perpendicular to their line of intersection.
In pyramids, the angle between a sloping edge and the base is found using the vertical height and half the base diagonal.
Example: In a square-based pyramid with side 8 cm and vertical height 10 cm, the angle between VA and base is tan⁻¹(10 / (8√2/2)).

Right triangle for a pyramid's sloping edge

Solving Multi-Step Problems

Break the problem into 2D right-angled triangles; solve each step sequentially.
Redraw each triangle flat on paper with all known lengths.
Use Pythagoras to find missing sides, then trigonometry to find angles.
If stuck, start finding any lengths or angles – they may lead to the answer and earn partial marks.

Triangular prism problem

Common Question Types

Finding the length of a space diagonal (e.g., longest object that fits in a box).
Calculating the angle between a diagonal and a face (e.g., angle between AG and base ABCD).
Finding the angle between a sloping edge and the base in a pyramid.
Using given ratios or volumes to find missing dimensions and then angles.

Cuboid box for a diagonal problem

Examiner Tips

The 3D Pythagoras formula is not provided – you must derive it or use 2D steps.
Always redraw 2D triangles flat to avoid misinterpreting the 3D diagram.
If unsure, start calculating any lengths or angles – you may score method marks.
Label all known lengths on the diagram before starting calculations.

Slides

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Practice questions

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1.In a cuboid with dimensions 3 cm, 4 cm and 6 cm, what is the length of the longest diagonal?
Easy
- A7.81 cm
- B9.22 cm
- C6.71 cm
- D13.0 cm
2.A cuboid has dimensions 5 cm, 12 cm and 13 cm. Find the angle between the longest diagonal and the base.
Medium
- A45.0°
- B30.0°
- C60.0°
- D22.6°
3.In a cuboid, the length of the diagonal of the base is 10 cm and the height is 6 cm. What is the length of the space diagonal?
Medium
- A11.66 cm
- B16.0 cm
- C8.0 cm
- D11.0 cm
4.A cuboid has dimensions 5 cm, 8 cm and 10 cm. Calculate the angle between the space diagonal and the base.
Medium
- A46.7°
- B43.8°
- C52.4°
- D37.6°
5.In the cuboid ABCDEFGH, AB = 6 cm, BC = 8 cm and CG = 10 cm. Find the length of AG.
Medium
- A14.1 cm
- B10.0 cm
- C12.8 cm
- D16.0 cm
6.A cuboid has dimensions 3 cm, 4 cm and 12 cm. What is the angle between the space diagonal and the longest edge?
Medium
- A22.6°
- B67.4°
- C45.0°
- D30.0°
7.A pyramid has a square base of side 6 cm and a vertical height of 8 cm. Find the angle between a sloping edge and the base.
Medium
- A62.0°
- B28.0°
- C45.0°
- D55.0°
8.A cuboid has dimensions 2 cm, 3 cm and 6 cm. What is the length of the space diagonal?
Easy
- A7.0 cm
- B6.0 cm
- C7.81 cm
- D6.71 cm

More questions

9.In a cuboid, the space diagonal is 13 cm and the base diagonal is 12 cm. What is the height?
Medium
- A5 cm
- B1 cm
- C25 cm
- D7 cm
10.A cuboid has dimensions 9 cm, 12 cm and 20 cm. Find the angle between the space diagonal and the base.
Medium
- A50.2°
- B39.8°
- C53.1°
- D36.9°
11.A pyramid has a square base of side 8 cm. The sloping edges are each 12 cm long. Find the vertical height of the pyramid.
Hard
- A10.6 cm
- B8.94 cm
- C11.3 cm
- D9.80 cm
12.A cuboid has dimensions 5 cm, 5 cm and 10 cm. What is the angle between the space diagonal and the base?
Medium
- A54.7°
- B35.3°
- C45.0°
- D60.0°
13.In a cuboid, the length, width and height are 3 cm, 4 cm and 5 cm. What is the length of the space diagonal?
Easy
- A7.07 cm
- B6.0 cm
- C8.60 cm
- D5.0 cm
14.A cuboid has dimensions 6 cm, 8 cm and 10 cm. What is the length of the space diagonal?
Easy
- A\√{200} cm
- B\√{224} cm
- C\√{2000} cm
- D24 cm
15.In a cuboid, the base diagonal is 10 cm and the space diagonal is 15 cm. What is the height of the cuboid?
Medium
- A5 cm
- B\√{125} cm
- C\√{325} cm
- D25 cm
16.A cuboid has dimensions 4 cm, 7 cm and 9 cm. Calculate the angle between the space diagonal and the base.
Hard
- A\tan^-1(\frac{9}{\√{65}})
- B\tan^-1(\frac{\√{65}}{9})
- C\sin^-1(\frac{9}{\√{146}})
- D\cos^-1(\frac{4}{\√{146}})
17.A pyramid has a square base of side 10 cm and a vertical height of 12 cm. Find the angle between a sloping edge and the base.
Hard
- A\tan^-1(\frac{12}{5\√{2}})
- B\tan^-1(\frac{5\√{2}}{12})
- C\sin^-1(\frac{12}{10})
- D\cos^-1(\frac{5}{12})
18.A cuboid has dimensions 1 cm, 2 cm and 2 cm. Find the length of the space diagonal.
Easy
- A3 cm
- B\√{9} cm
- C\√{5} cm
- D\√{6} cm
19.What is the 3D version of Pythagoras' theorem for a cuboid with dimensions x, y, and z?
Easy
- Ad² = x² + y² + z²
- Bd² = x² + y² - z²
- Cd = x + y + z
- Dd² = (x + y + z)²
20.In a cuboid, which of the following is the correct method to find the angle between a line and a plane?
Medium
- AForm a right-angled triangle with the line as the hypotenuse and the perpendicular height to the plane as one side.
- BUse the dot product of the line vector and the plane normal.
- CMeasure the angle between the line and any line in the plane.
- DUse the law of cosines on the triangle formed by the line and two points on the plane.
21.A cuboid has dimensions 3 cm, 4 cm, and 12 cm. What is the length of the space diagonal?
Easy
- A13 cm
- B14 cm
- C15 cm
- D16 cm
22.A cuboid has dimensions 6 cm, 8 cm, and 10 cm. Find the angle between the space diagonal and the base.
Medium
- A45°
- B30°
- C60°
- D53°
23.A pyramid has a square base of side 8 cm and a vertical height of 10 cm. Find the angle between a sloping edge and the base (to the nearest degree).
Medium
- A61°
- B51°
- C45°
- D39°
24.A cuboid has dimensions 4 cm, 7 cm, and 9 cm. Calculate the angle between the space diagonal and the base. Give your answer correct to one decimal place.
Medium
- A48.1°
- B41.9°
- C35.4°
- D19.3°
25.A cuboid has dimensions 5 cm, 12 cm, and 13 cm. Find the angle between the space diagonal and the base.
Medium
- A45°
- B30°
- C60°
- D50°
26.A pyramid has a square base of side 10 cm and a vertical height of 12 cm. Find the angle between a sloping edge and the base. Give your answer correct to the nearest degree.
Medium
- A59°
- B65°
- C55°
- D50°
27.In a cuboid, the space diagonal is the line joining two opposite vertices. Which of the following is NOT a space diagonal?
Easy
- AAG
- BBH
- CCE
- DAB
28.In 3D, you can always split a problem into two 2D right-angled triangles.
Medium
True or false?
29.The 3D Pythagoras formula is provided in the exam formula sheet.
Easy
True or false?
30.To find the angle between a line and a plane, you form a right-angled triangle where the height is:
Easy
- Aperpendicular to the plane
- Bparallel to the plane
- Calong the line
- Dalong the base of the plane
31.In a cuboid, the space diagonal joins two opposite vertices. How many space diagonals does a cuboid have?
Easy
- A4
- B2
- C8
- D6
32.When finding the angle between a line and a plane, you can use any right-angled triangle that includes the line and a point on the plane.
Medium
True or false?
33.A pyramid has a square base of side 8 cm and a vertical height of 10 cm. Which of the following is the correct expression for the length of a sloping edge?
Medium
- A\√{8² + 10²}
- B\√{(4\√{2})² + 10²}
- C\√{8² + 8² + 10²}
- D\√{4² + 10²}
34.Match each 3D shape with the correct method to find the length of the space diagonal.
Medium
- Cuboid with dimensions a, b, c
- Cube of side s
- Square-based pyramid of base side s and height h
- d = √(a² + b² + c²)
- d = s√3
- d = √(s²/2 + h²)
35.Arrange the steps in the correct order to find the angle between the space diagonal and the base of a cuboid with dimensions 3 cm, 4 cm, and 6 cm.
Medium
- Calculate the base diagonal using Pythagoras.
- Form a right-angled triangle using the height and the base diagonal.
- Use tan(θ) = height / base diagonal.
- Find θ = arctan(height / base diagonal).
36.Which of the following statements about 3D Pythagoras and trigonometry are correct? (Select all that apply)
Hard
- AThe 3D Pythagoras formula is given in the exam formula sheet.
- BYou can always split a 3D problem into two 2D right-angled triangles.
- CThe angle between a line and a plane is found using a right-angled triangle where the height is perpendicular to the plane.
- DIn a cuboid, the space diagonal is the longest distance between any two vertices.
37.In a cuboid, the space diagonal can be found using the formula d² = x² + y² + z² where x, y, z are the dimensions.
Easy
True or false?
38.A cuboid has dimensions 3 cm, 4 cm and 6 cm. Calculate the length of the space diagonal. Give your answer correct to 3 significant figures.
Medium
39.Name the method used to find missing sides or angles in right-angled triangles in 3D problems.
Easy
40.Complete the sentence.
Medium
To find the angle between a line and a plane, first identify the ____ and the ____, then form a right-angled triangle.
41.A cuboid has dimensions 5 cm, 8 cm and 10 cm. Find the length of the space diagonal. Give your answer correct to 3 significant figures.
Medium
42.A cuboid has dimensions 2 cm, 3 cm, and 6 cm. Calculate the length of the space diagonal. Give your answer correct to 3 significant figures.
Medium
43.Complete the sentence: To find the angle between a line and a plane, you form a right-angled triangle where the height is ___ to the plane.
Easy
To find the angle between a line and a plane, you form a right-angled triangle where the height is ____ to the plane.
44.A pyramid has a square base of side 8 cm and a vertical height of 10 cm. Find the angle between a sloping edge and the base. Give your answer correct to 1 decimal place.
Medium
45.Explain how you would find the angle between the space diagonal and the base of a cuboid with dimensions 3 cm, 4 cm and 6 cm.
Medium
46.A cuboid has dimensions 3 cm, 4 cm and 6 cm. Calculate the length of the space diagonal. Give your answer correct to 3 significant figures.
Medium
- A6.81 cm
- B9.81 cm
- C7.81 cm
- D8.81 cm
47.A cuboid has dimensions 5 cm, 8 cm and 10 cm. Find the length of the space diagonal. Give your answer correct to 3 significant figures.
Medium
- A15.7 cm
- B14.7 cm
- C12.7 cm
- D13.7 cm
48.A cuboid has dimensions 2 cm, 3 cm, and 6 cm. Calculate the length of the space diagonal. Give your answer correct to 3 significant figures.
Easy
- A7 cm
- B6 cm
- C8 cm
- D9 cm

Revision notes

## 3D Pythagoras & Trigonometry ### Key Formulas - **3D Pythagoras**: \(d^2 = x^2 + y^2 + z^2\) (distance between two points in 3D) - **SOHCAHTOA**: \(\sin\theta = \frac{\text{opp}}{\text{hyp}},\; \cos\theta = \frac{\text{adj}}{\text{hyp}},\; \tan\theta = \frac{\text{opp}}{\text{adj}}\) ### Angle Between a Line and a Plane 1. Identify the plane and the line. 2. Drop a perpendicular from the line to the plane to form a right-angled triangle. 3. Use SOHCAHTOA to find the angle. ### Problem-Solving Strategy - **Split 3D into 2D**: Break the problem into two right-angled triangles in 2D. - **Redraw triangles flat**: Draw 2D triangles on paper (not in perspective) to apply Pythagoras or trig. - **Start anywhere**: If stuck, find any lengths/angles—they may help later. ### Common Shapes - **Cuboid**: Diagonals use 3D Pythagoras. Angle between a space diagonal and a face uses trig. - **Pyramid**: Base is square; apex above centre. Use right triangles from apex to base centre. - **Prism**: Cross-section is a triangle. Use right triangles within the prism. ### Tips - You are **not** given the 3D Pythagoras formula in the exam—memorise it. - Always look for right angles in the 3D shape. - For angle between line and plane, imagine lowering a fishing line vertically onto the plane.

3D Pythagoras: A formula to find the distance between two points in three dimensions: d² = x² + y² + z².
Angle between a line and a plane: The acute angle between the line and its projection onto the plane, found using a right-angled triangle with the perpendicular height.
SOHCAHTOA: A mnemonic for trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

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