// maths › Right-Angle Trigonometry
Solve three-dimensional problems on a cuboid step by step: find the base diagonal, then the space diagonal, then the angle it makes with the base, with a wireframe box that highlights each right-angled triangle in turn.
diagonal = √(l² + w² + h²) · angle = tan⁻¹(h / √(l²+w²))
Break it into right-angled triangles you can see flat. For a cuboid, first find the base diagonal with Pythagoras, then use that diagonal and the height to find the space diagonal, then the angle to the base.
It is the longest straight line inside the box, from one corner to the opposite corner. Its length is √(length² + width² + height²).
For a box 3 by 4 by 12, the base diagonal is √(3² + 4²) = 5, the space diagonal is √(5² + 12²) = 13, and the angle it makes with the base is tan⁻¹(12 ÷ 5) ≈ 67.4°.
A 3D line cannot be solved in one step, but it can be seen as the hypotenuse of an upright right triangle whose base is a diagonal lying flat — so two ordinary 2D steps solve it.
Architects and structural engineers size diagonal braces and roof members, packaging designers check whether items fit in boxes, pilots and air-traffic controllers reason about 3D paths, and 3D modellers and game developers compute spatial distances.