3D geometry — volume, surface area, and packaging (IB Math AI SL)
Updated October 2026
Quick answer: Volumes and surface areas of prisms, cylinders, pyramids, cones and spheres are all in the formula booklet; the skill is splitting a solid into these shapes. For a cone you often need the slant height l = √(r² + h²) first, because the curved surface area is πrl.
SL AI's geometry is applied: which cereal box has the lowest surface-area-to-volume ratio? Master the volume + surface-area formulas for cylinders, cones, and truncated shapes.
Practise 3D geometry — volume, surface area, and packaging 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.
Volume of a prism or cylinder = area of cross-section × length; a pyramid or cone is one third of that.
Sphere: V = (4/3)πr³ and A = 4πr². A hemisphere has half the volume and half the curved area, plus a flat circle if the base is exposed.
Cone: curved surface area = πrl, where l is the slant height, not the vertical height.
For a compound solid, add the volumes, but for the surface area add only the faces you can see: a joined face is inside the solid.
Keep units consistent: cm³ for volume, cm² for area, and 1 litre = 1000 cm³.
Worked example
A toy is a cone of base radius 3 cm and height 10 cm with a hemisphere of radius 3 cm fixed to its base. Find the volume and the total surface area of the toy.
The curved surface area is πrl, where r is the base radius and l the slant height; if you are given the vertical height h, find l = √(r² + h²) first. Add πr² for the base only if it is part of the outside of the shape.
Where can I practise 3D geometry questions?
On the 3D Coordinate Geometry 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 geometry?
Yes: a GDC is allowed on every SL AI paper, so practise the calculator method as well as the working.