Sheet № 163 · Higher only · AQA · Edexcel · OCR
Pythagoras in 3D –
Pythagoras in 3D extends the familiar a² + b² = c² into three dimensions. It is a Higher-only GCSE Maths topic tested by AQA, Edexcel, and OCR. Questions typically ask you to find the space diagonal of a cuboid, the longest rod that fits inside a box, or a slant length in a pyramid. The key skill is identifying the right-angled triangle h
§Key definitions
Question:
A cuboid has dimensions 3 cm by 4 cm by 12 cm. Find the length of the space diagonal.
Answer:
The space diagonal is 13 cm.
Q1 (Foundation):
This topic is Higher only.
Q2 (Higher):
A cuboid has dimensions 5 cm, 12 cm, and 8 cm. Find the space diagonal to 1 d.p.
Q3 (Higher):
A square-based pyramid has a base of 10 cm and a vertical height of 12 cm. Find the slant height to 1 d.p.
§Formulas to memorise
For a cuboid with sides a, b, and c — space diagonal d = sqrt(a² + b² + c²)
Pythagoras' theorem: a² + b² = c² (used twice in 3D problems)
Draw or label the 3D shape — with all given dimensions.
Identify the right-angled triangle on the base. — Use Pythagoras to find the base diagonal.
Form a second right-angled triangle — using the base diagonal and the height of the shape.
Apply Pythagoras again — to find the space diagonal or the required length.
Round — to the required degree of accuracy.
Worked example
A cuboid has dimensions 3 cm by 4 cm by 12 cm. Find the length of the space diagonal.
This topic is Higher only, but this example uses simpler numbers.
⚠ Common mistakes
- ✗Applying Pythagoras only once. 3D problems almost always require two applications of Pythagoras. Students who stop after the first triangle miss the final answer.
- ✗Using the wrong lengths in the triangle. Carefully identify which lengths form the right-angled triangle. Sketch the triangle separately from the 3D shape to avoid confusion.
- ✗Confusing the slant height with the vertical height. The slant height runs along the face of a pyramid; the vertical height runs straight up from the base to the apex. They are different measurements.
✦ Exam tips
- →Always sketch the right-angled triangle you are working with separately — this prevents errors from misreading the 3D diagram.
- →The formula d = sqrt(a² + b² + c²) is a shortcut for cuboids, but make sure you understand why it works (two applications of Pythagoras).
- →"The longest rod that fits inside a box" means the space diagonal — this is a common exam phrasing.
- →Leave your intermediate answers unrounded. Only round at the final step to avoid cumulative rounding errors.