3D Pythagoras and trigonometry
Pythagoras and SOHCAHTOA in 3D shapes — cuboids, pyramids, prisms.
Higher-tier geometry: find a 2D right triangle inside the 3D shape, apply Pythagoras or SOHCAHTOA to it, then to a second triangle if needed.
Worked examples
Cuboid 3×4×12: space diagonal √(3²+4²+12²) = 13.
Square pyramid base 6, vertical height 4: slant edge = √((3√2)² + 4²) = √34.
Angle a space diagonal makes with the base: arctan(height / face diagonal).
Frequently asked questions
Method?
Always sketch and label. Pick a 2D triangle inside the 3D shape (often using a face diagonal or vertical line).
Where does 3D Pythagoras come from?
Apply Pythagoras twice: face diagonal first, then with the height.