Further Pythagoras' Theorem: 3D Problems and Space Diagonals
Apply Pythagoras' theorem in 3D for IB MYP Extended Year 5. Learn space diagonals of cuboids, compound 3D problems, and avoid the most common mistakes.
Moving from 2D to 3D
In Standard level, Pythagoras' theorem is applied to flat (2D) triangles. In the Extended pathway, you apply the same theorem twice — once within a face of a 3D shape, and once across the full three-dimensional space — to find lengths that cannot be measured directly.
Space Diagonal of a Cuboid
The most common 3D Pythagoras problem involves a cuboid with dimensions l, w, and h. The space diagonal d (the diagonal running from one corner of the cuboid to the opposite corner through the interior) is found using:
d² = l² + w² + h²
This formula is derived by applying Pythagoras twice: first to find the base diagonal, then to include the height.
Compound 3D Problems
Some questions embed a 3D Pythagoras step inside a larger problem. For example, you might need to find the slant height of a pyramid, the diagonal of a triangular prism, or the length of a wire stretching from a corner of a room to a point on the opposite wall. Always draw a clear 2D cross-section or working triangle and label the right angle explicitly.
Key Difference from Standard Level
The challenge is not the formula — it is identifying which right-angled triangle to construct inside the 3D figure. This spatial reasoning step is where most marks are won or lost.
Common Mistakes
- Forgetting to take the square root at the end of each Pythagoras step
- Using the full space diagonal formula when the question only asks for a face diagonal
- Drawing an incorrect cross-section that does not contain a right angle
MYP Question Style
Questions are usually set in practical contexts — rooms, boxes, ramps, or outdoor structures. Criterion D tasks may ask you to justify your method and interpret the answer in context, not just calculate.