3D trigonometry — angle of elevation and depression (IB Math AI SL)
Updated October 2026
Quick answer: Find a right-angled triangle that contains the length or angle you want, draw it flat on its own, and use Pythagoras or SOH CAH TOA. The angle between a line and a plane is the angle between the line and its shadow on the plane.
Standing at the top of a cliff looking down at a boat, or standing at the base of a tower looking up. SL AI always asks for BOTH the horizontal distance and the vertical height.
Practise 3D trigonometry — angle of elevation and depression with AI marking
Open the full SL AI engine — every question in this unit is worked through with instant AI-graded feedback on your handwritten workings.
Every 3D question reduces to one or two right-angled triangles. Sketch each one flat and label it before you calculate.
Use the base first: find the diagonal across the base, then use it with the height.
The distance between (x₁, y₁, z₁) and (x₂, y₂, z₂) is √((x₁ − x₂)² + (y₁ − y₂)² + (z₁ − z₂)²).
The angle between a line and a plane is the angle between the line and its projection on the plane (the point directly below the top of the line).
Angles of elevation and depression are measured from the horizontal, never from the vertical.
Worked example
A cuboid ABCDEFGH has a base ABCD that is 8 cm by 6 cm and a height of 5 cm. G is vertically above C. Find the length of the diagonal AG and the angle between AG and the base.
Base diagonal AC = √(8² + 6²) = 10 cm.
Triangle ACG has a right angle at C, with CG = 5 cm.
AG = √(10² + 5²) = √125 = 11.2 cm (3 s.f.).
The projection of AG on the base is AC, so the angle is tan⁻¹(5 ÷ 10) = 26.6° (3 s.f.).
How do you find the angle between a line and a plane?
Drop a perpendicular from the top of the line to the plane to find its projection. The angle you want is in the right-angled triangle formed by the line, its projection and the perpendicular.
Where can I practise 3D trigonometry questions?
On the 3D Trigonometry questions page: 13 exam-style SL AI questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for 3D trigonometry?
Yes: a GDC is allowed on every SL AI paper, so practise the calculator method as well as the working.