3D Pythagoras and Trigonometry

Reduce it to two dimensions

• Nothing new is needed here: the tools are still Pythagoras' theorem and SOHCAHTOA
• The whole difficulty is finding the right-angled triangle inside the solid, so the first move is always to locate one
• Once found, redraw that triangle flat on the page rather than leaving it drawn at an angle inside the solid

• A perspective drawing distorts every angle, so a triangle that looks right-angled in the picture may not be, and one that looks skewed often is
• Many problems need two triangles in succession, where the answer from the first becomes a side of the second
• Keep the full unrounded value between the two stages and round only at the very end

The space diagonal

• The longest straight line inside a cuboid runs between opposite corners, and it is what a "longest rod that fits in the box" question asks for
• Reach it in two steps: Pythagoras across the base to get the base diagonal, then Pythagoras again using that diagonal and the height

• There is a one-step version, d² = x² + y² + z², but it is not printed in the paper, so the two-step route is the reliable one
• The two-step route also shows the working the marks are awarded for