3D Geometry (AI HL)
Three-dimensional problems look intimidating but almost always reduce to something familiar: a distance formula with an extra coordinate, a standard solid's volume or surface area, or a right-angled triangle hiding inside a box, cone or pyramid. This page covers the skills specific to solids and space, with worked examples and the traps that catch students under exam pressure. It's part of the broader Geometry & Trigonometry topic.
11 questions on this sub-topic.
The key tools
Covered under IB syllabus reference SL3.1: the distance between two points in 3D and their midpoint, volume and surface area of standard solids, and the angle between two lines or between a line and a plane.
3D distance
\(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\)
The 2D distance formula with a third coordinate added. The midpoint formula extends the same way, averaging each coordinate.
Volume and surface area
Cones, spheres, hemispheres, pyramids and combinations of these are all on the formula booklet - non-standard solids like a frustum have their formula given in the question.
✓ Standard solids in the formula bookletWant the full topic overview and syllabus table, plus GDC guidance for these calculations? See Geometry & Trigonometry (including using your GDC).
Worked examples
A cone has radius 6 cm and height 10 cm.
Find its volume (3 significant figures).
Worked solution
\(V = \tfrac13 \pi r^2 h = \tfrac13 \pi(36)(10)\) M1
\(= 120\pi \approx 377\) cm³. A1
A square-based pyramid has base side 10 cm and apex 12 cm directly above the centre.
Find the angle between a triangular face and the base (3 significant figures).
Worked solution
Distance from centre to base-edge midpoint \(= 5.\) M1
\(\tan\theta = \dfrac{12}{5}\) M1 A1
\(\theta = 67.4^{\circ}.\) A1
A bucket is a frustum with top radius 12 cm, base radius 8 cm, height 20 cm.
Find its volume (3 significant figures).
Worked solution
\(V = \tfrac13 \pi h (R^2 + Rr + r^2)\) M1
\(= \tfrac13 \pi(20)(144 + 96 + 64)\) A1
\(= \tfrac{20\pi}{3}(304)\) A1
\(\approx 6367\) cm³. A1
Common mistakes
- Dropping a coordinate in the distance formula. With three coordinates instead of two it's easy to only square and add two of the three differences - always check all of \(x\), \(y\) and \(z\) are accounted for.
- Not isolating the right-angled triangle. The angle between a line and a plane, or between a face and a base, only becomes solvable once you find the correct perpendicular height and the correct base length - sketch the triangle separately if it helps.
- Using diameter instead of radius. Cone, sphere and hemisphere formulas all use \(r\), not the diameter - halve any diameter given in the question before substituting.
Ready to practise properly?
11 3D geometry questions, marked instantly like the real exam.
Quick answers
How do you find the distance between two points in 3D?
\(d = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}\) - the usual 2D distance formula with a third coordinate added.
How do you find the angle between a line and a plane?
Identify the right-angled triangle formed by the line, its projection onto the plane, and the perpendicular height between them, then use basic trig (usually \(\tan\)) to find the angle.