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Playbook

Mensuration 3D

171 q · 17% HARD · falling from 8.1 to 6.7 a paper. Eight of ten subtopics are under 20% HARD; solids inside solids (53%) is the one page to attempt only when the set-up is clear.

Questions in the bank
171
q/paper (21 papers)
8.14
Tagged HARD
17%
Subtopics
10

Strand: Cornerstone

When you’ll see it

Volume or surface area of a solid, melting one solid into others, or water rising in a vessel.

How this chapter is tested

171 questions, falling from 8.1 a paper in 2016-2020 to 6.7 in 2024-2026. Eight of its ten subtopics are under 20% HARD: it is a formula chapter with one expensive page.

Solids inside solids (15 questions, 53% HARD) — the largest cone in a cube, the sphere inside a cylinder — is the pool. Scaling (32%) asks how volume changes when every length changes; the answer is the cube of the length factor.

The sub-skills

The distinct skills inside the chapter, in the order to learn them.

  • Cuboids and cubes

    Volume, surface area and the space diagonal √(l² + b² + h²).

  • Cylinders, cones and spheres

    Curved and total surface areas; volumes πr²h, ⅓πr²h, ⁴⁄₃πr³.

  • Melting and recasting

    Volume is conserved; count pieces as total volume ÷ one piece.

  • Water displacement

    The rise in level × base area = volume immersed.

  • Scaling

    Lengths × k means areas × k² and volumes × k³.

Traps to expect

Distractor shapes this chapter reuses. The Traps page covers the ones that cut across chapters.

  • Curved or total surface area

    A cylinder's total surface adds 2πr² to the curved 2πrh.

  • Hollow or solid hemisphere

    A solid hemisphere's surface is 3πr², not 2πr².

  • Doubling the radius doubles the volume

    It multiplies it by 8 for a sphere and by 4 for a cylinder of the same height.

Learn it before you drill it

This chapter has full teaching notes — foundations, worked examples, self-checks and a mastery check for each subtopic. Read the notes once, then drill subtopic by subtopic below.

Mensuration 3D notes

Drill every Mensuration 3D question

171 questions from the bank, across 10 subtopics.

Drill the 171 questions

Drill one subtopic at a time

The 10 subtopics, in teaching order.

Related playbooks

Often paired with this one — the technique or the trap overlaps. Drill these next.