3D Pythagoras
Grade 6The big idea
Pythagoras' theorem still works in 3D — you just apply it in stages. Find a diagonal on a face first (a 2D Pythagoras step), then use that diagonal as a side for the 3D step.
The most common question is finding the length of the space diagonal of a cuboid — the diagonal that cuts corner to corner through the inside of the box.
The space diagonal formula
For a cuboid with length l, width w, and height h:
d² = l² + w² + h²
This comes from two Pythagoras steps: first d₁² = l² + w² for the base diagonal, then d² = d₁² + h². Combining gives the three-term version above.
Worked example
EXAMPLE
A cuboid measures 3 cm × 4 cm × 12 cm. Find the length of the space diagonal.
d² = 3² + 4² + 12² = 9 + 16 + 144 = 169. d = √169 = 13 cm.
Always draw a sketch
Draw the 3D shape and mark the diagonal clearly. Identifying which right-angled triangle to use is the hardest part — the calculation itself is straightforward once you can see it.
