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:
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
Always draw a sketch
Try this one
A cuboid box has a base measuring \( 8 \) cm by \( 6 \) cm and a height of \( 5 \) cm. A drinking straw is placed inside so that it runs from one bottom corner to the top corner diagonally opposite.
Calculate the angle that the straw makes with the base of the box. Give your answer to \( 1 \) decimal place. [4]
You need the base diagonal first, then trigonometry in an upright triangle.
Ask yourself: 'which length is opposite the angle and which is adjacent to it?' The height rises above the base diagonal, so once both are known you can use the tangent ratio.
