Angles in 3-D figures: what the diagram is not telling you

6 min read·Updated 5 September 2026

Every 3-D figure you meet in a Paper 2 question is a flat drawing of a solid object, and flattening destroys information. Lengths that are equal in the solid are drawn unequal; angles that are right are drawn as anything but; angles that are nothing like right are drawn looking square. A 3-D question is therefore never about reading the picture — it is about finding a triangle you can be sure is right-angled, and doing ordinary 2-D trigonometry in it.

The skill being tested is that one move: pick the right triangle out of the solid, then forget the solid exists.

Misconception 1: trusting a right angle you were not given

A vertical edge of a cuboid genuinely is perpendicular to the base, because that is what a cuboid is. A diagonal of a face is genuinely not. In the drawing they can look identical. Students apply Pythagoras or a sine ratio to a triangle that simply is not right-angled, and every step afterwards is fiction.

Before using any right-angle fact, name the reason: "this edge is perpendicular to that plane because the solid is a cuboid", or "the apex is directly above the centre because the pyramid is described as right". If you cannot name a reason, you do not have a right angle.

Misconception 2: the angle between a line and a plane

This angle is defined by a projection: drop the line's endpoint straight down onto the plane, join the foot to the point where the line meets the plane, and the angle between the original line and that projection is the one wanted.

The common error is to use a convenient edge instead of the projection. In a box, the shadow of a space diagonal on the base is the base's diagonal, not one of the base's sides — and a side is right there, already drawn, already labelled with a length you were given. Using it produces a plausible angle that is simply a different angle.

Misconception 3: the angle between two planes

For two planes meeting along a common edge, choose a point on that edge and draw one line in each plane through that point, each perpendicular to the edge. The angle between those two lines is the angle between the planes.

The perpendicularity condition is the whole definition, and it is what gets skipped. Any two lines drawn "across" the two faces will produce an angle, and it will look reasonable, and it will be wrong. If your two arms are not both at right angles to the shared edge, you have measured something else.

A worked example

A cuboid has a rectangular base measuring $4$ cm by $3$ cm and a vertical height of $12$ cm. Find the angle between a space diagonal and the base, to the nearest degree.

First the projection. The space diagonal runs from a bottom corner to the opposite top corner; its shadow on the base is the base's diagonal, whose length is

The vertical edge is perpendicular to the base because the solid is a cuboid — that is the named reason — so the triangle made by the base diagonal, the vertical edge and the space diagonal is right-angled at the base. Hence

The wrong turn is to use an edge of the base instead of its diagonal: $\tan\theta = \tfrac{12}{4}$ gives about $72^\circ$, and $\tan\theta = \tfrac{12}{3}$ gives about $76^\circ$. All three are angles that genuinely exist in this box. Only one of them is the angle between the space diagonal and the base.

Check yourself

Say the sentence before you compute: "the triangle I am using is right-angled at ___, because ___." If the blank after "because" needs the diagram rather than a property of the solid, stop and find a different triangle.

The drill below regenerates the solid and its dimensions every time, so the projection has to be worked out afresh instead of remembered.

Original study notes by Dsetopia, written from the EDB syllabus learning objectives. Not affiliated with the HKEAA; no exam content is reproduced.