3D Trigonometry

Reveal the right-angled triangles inside a cuboid and use Pythagoras and trigonometry to solve them.

Curriculum rangeYears 9–11

Length l = 4Width w = 5Height h = 3

Base diagonal (2D Pythagoras across the base rectangle)

p = \sqrt{l^2+w^2} = \sqrt{4^2+5^2} = 6.4

Space diagonal (3D Pythagoras, using p and h)

d = \sqrt{p^2+h^2} = \sqrt{6.4^2+3^2} = 7.1

Angle between the diagonal and the base

\tan\theta = \dfrac{h}{p} = \dfrac{3}{6.4} \Rightarrow \theta = 25.1°

The pink dashed line is the diagonal across the base -- it isn't a real edge of the cuboid, it's the "shadow" of the space diagonal on the base. That turns the space diagonal into the hypotenuse of a right-angled triangle you can solve with normal Pythagoras and trigonometry.