GCSE MATHS • HIGHER TIER

3D Pythagoras Made Simple

Use Pythagoras twice to find face diagonals and distances through three-dimensional shapes.

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3D Pythagoras in Four Moves

1

Draw

Sketch the 3D shape and label the known lengths.

2

Find the Face Diagonal

Identify a right-angled triangle on one face and apply Pythagoras.

3

Apply Pythagoras Again

Use the face diagonal with the third dimension to form a second right-angled triangle.

4

Check

Square root the final value, include units and round appropriately if required.

What You’ll Learn

Understand what 3D Pythagoras means.

Recognise when two applications of Pythagoras are needed.

Find a face diagonal in a cuboid.

Find a space diagonal through a 3D shape.

Use Pythagoras with 3D coordinates.

Avoid common Higher Tier exam mistakes.

Key Rules

Direct proportion means two quantities change at the same rate.

If one quantity doubles, the other quantity doubles.

If one quantity halves, the other quantity halves.

Write direct proportion as :

y∝x

Worked Proof: 3D Pythagoras

Step-by-Step Method

  • 1. Identify how many times the event occurred.
  • 2. Identify the total number of trials.
  • 3. Use the formula :
    Relative frequency = Event occurrences ÷ Total trials.
  • 4. Complete the division and write the answer as a decimal.
  • 5. For a prediction, multiply the relative frequency by the new number of trials.

More Worked Examples

1

A cuboid has dimensions 3 cm, 4 cm and 12 cm.

Step 1: Find the diagonal of the base.

3² + 4² = 25
Diagonal = 5 cm.
Step 2: Use the diagonal and height.
5² + 12² = 169
Distance = 13 cm.
Answer: 13 cm

2

A cuboid measures 6 cm, 8 cm and 24 cm.

Base diagonal :

6² + 8² = 100
Diagonal = 10 cm.
Space diagonal:
10² + 24² = 676
Distance = 26 cm.
Answer: 26 cm

3

A cuboid measures 5 cm, 12 cm and 9 cm.

Base diagonal :

5² + 12² = 169
Diagonal = 13 cm.
Space diagonal:
13² + 9² = 250
Distance = √250 ≈ 15.8 cm.
Answer: 15.8 cm

4

A cuboid measures 7 cm, 24 cm and 10 cm.

Base diagonal :

7² + 24² = 625
Diagonal = 25 cm.
Space diagonal :
25² + 10² = 725
Distance ≈ 26.9 cm.
Answer: 26.9 cm
  • Common GCSE Mistakes
  • Using Pythagoras only once.
  • Choosing the wrong diagonal.
  • Forgetting to square root the final answer.
  • Making arithmetic errors.
  • Making arithmetic errors.
  • Exam Tips
  • Draw and label the 3D shape before calculating.
  • Find the face diagonal before the space diagonal.
  • Check that each triangle you use is right-angled.
  • Keep exact values until the final step where possible.
  • Remember to square root after finding the squared distance.
  • Include units and round only when the question requires it.

Skills Used in 3D Pythagoras

Pythagoras' Theorem

Right-Angled Triangles

3D Shapes

Face Diagonals

Space Diagonals

Square Roots

3D Coordinates

Geometry

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Step-by-step walkthroughs of 3D Pythagoras with clear methods and exam tips.

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Frequently Asked Questions

It extends Pythagoras' Theorem into three dimensions and is usually used twice to find a distance through a 3D shape.

The first application finds a face diagonal; that diagonal is then used in a second right-angled triangle to find the overall distance.

It is the distance between two opposite corners through a three-dimensional shape such as a cuboid.

Use it when a required distance passes through a 3D shape and two right-angled triangles can be formed.

Yes. The source states that the same principle can be applied repeatedly to distances in three-dimensional coordinates.

The source recommends SOHCAHTOA, Exact Trig Values, Sine Rule, Cosine Rule, Bearings and Trigonometry in 3D.