CAT Quant · Study & Practice

3D Mensuration

AreaMensuration DifficultyModerate CAT weightage1–3 questions (Geometry–Mensuration block, often inside DI-style or mixed-solid problems)

3D mensuration is the study of volume and surface area of solids — cubes, cuboids, cylinders, cones, spheres, hemispheres and frustums. In CAT, XAT and SNAP this topic rewards the student who has the formulas at fingertips and, more importantly, knows when to subtract one solid from another, when a shape is melted and recast (volume is conserved, surface area is not), and when only the curved or lateral part matters. The questions are rarely plug-and-play: a typical CAT setup hollows out a cylinder, inscribes a sphere in a cube, recasts a cone into spheres, or paints only the visible faces of a stack of cubes. Three ideas unlock almost every problem here. First, melting/recasting keeps volume constant, so equate volumes. Second, total surface area is curved/lateral area plus the flat circular or polygonal ends — and you must decide which ends actually exist (a closed tank has a lid, an open one does not). Third, ratios scale predictably: if every linear dimension scales by k, area scales by k squared and volume by k cubed. This chapter covers each solid in turn with the volume formula, the surface-area split, the diagonal where relevant, and the CAT-style traps, so you can read a worded solid problem and write the right equation in one line.

Topics

1 Cube & Cuboid 4 worked · 5 practice · 8-Q test Start → 2 Cylinder & Cone 4 worked · 5 practice · 8-Q test Start → 3 Sphere & Hemisphere 4 worked · 5 practice · 8-Q test Start → 4 Frustum 4 worked · 5 practice · 8-Q test Start →

⚡ CAT shortcuts & speed methods

The fastest ways to crack this chapter under time pressure — the techniques that separate a 95+ percentiler from the rest.

  • Melting or recasting conserves VOLUME, not surface area — equate volumes to find the unknown dimension.
  • Linear scale k ⇒ area scales k², volume scales k³. Doubling an edge ⇒ 4× area, 8× volume.
  • Cone = exactly ⅓ of the cylinder on the same base and height; sphere = ⅔ of that cylinder when h = 2r.
  • Painted cube of side n: corners (3 faces) = 8, edges (2) = 12(n−2), faces (1) = 6(n−2)², hidden = (n−2)³.
  • Cone slant l = √(r²+h²); frustum slant l = √(h²+(R−r)²) — use the radius DIFFERENCE for slant.
  • Always ask: closed or open solid? Drop the missing circular/flat face before applying the TSA formula.

⚠️ Common mistakes & traps

CAT is designed so that careless errors here cost you marks. Internalise each trap before the exam.

  • Using vertical height h instead of slant height l in cone/frustum curved-surface-area formulas.
  • Adding the two circular ends to a cylinder/cone when the solid is open at one end (or vice versa).
  • Assuming surface area is conserved when a solid is melted and recast — only volume is conserved.
  • In the frustum slant, using (R + r) instead of (R − r): slant uses the difference, CSA uses the sum.
  • Confusing diameter and radius — halve the diameter before substituting into any volume or area formula.

📈 CAT exam insight & PYQ analysis

In CAT, XAT and SNAP, 3D mensuration appears as one or two questions, usually fused with recasting (a cone melted into spheres), composite solids (a cone on a hemisphere, a cylinder topped by a hemisphere), or inscribed-solid logic (largest sphere in a cube, cylinder in a cone). Pure formula recall is rare; the value lies in conserving volume and tracking which surfaces actually exist. Recurring patterns: hollow pipes and π(R²−r²)h, painted-and-cut cubes, and frustum buckets where slant height must be derived first. Prioritise the volume-conservation setups and the surface-area-of-composite-solid questions — they carry the marks and the traps.

🎴 Flashcards — instant recall

Tap a card to reveal the answer. Drill these until they are automatic.

Volume and TSA of a cube of side a?Tap to reveal
V = a³, TSA = 6a²
Space diagonal of a cube?Tap to reveal
a√3
Diagonal of a cuboid (l, b, h)?Tap to reveal
√(l²+b²+h²)
CSA and TSA of a cylinder?Tap to reveal
CSA = 2πrh, TSA = 2πr(r+h)
Volume of a cone?Tap to reveal
⅓πr²h
Slant height of a cone?Tap to reveal
l = √(r²+h²)
CSA and TSA of a cone?Tap to reveal
CSA = πrl, TSA = πr(r+l)
Volume and surface area of a sphere?Tap to reveal
V = (4/3)πr³, SA = 4πr²
CSA and TSA of a hemisphere?Tap to reveal
CSA = 2πr², TSA = 3πr²
Volume of a frustum (R, r, h)?Tap to reveal
(1/3)πh(R²+Rr+r²)
Slant height of a frustum?Tap to reveal
l = √(h²+(R−r)²)
Curved surface area of a frustum?Tap to reveal
πl(R+r)

📌 Quick revision

Lock in the six volume formulas (a³, lbh, πr²h, ⅓πr²h, (4/3)πr³, frustum (1/3)πh(R²+Rr+r²)) and their surface areas. Cube diagonal is a√3, cuboid diagonal √(l²+b²+h²). Cone slant l = √(r²+h²) and frustum slant l = √(h²+(R−r)²) — difference for slant, sum (R+r) for CSA. A cone is ⅓ of its cylinder; a hemisphere TSA is 3πr². When a solid is melted, equate volumes; when scaling, area goes as k² and volume as k³. Decide open versus closed before using any TSA, and never substitute slant for height or diameter for radius.

Chapter test

📝 3D Mensuration — Chapter Test
25 fresh questions · timed · full solutions
Start chapter test →
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🏆 Vidaara CAT success checklist

You have truly mastered 3D Mensuration when you can tick every box below.

  • Recall every formula in this chapter without looking them up
  • Solve each topic’s practice set with at least 80% accuracy
  • Use the chapter shortcuts to cut your solving time in half
  • Spot and avoid every common trap listed above
  • Score 80%+ on the timed chapter test

📋 Chapter mastery scorecard

Track where you stand. Aim for the target before moving to the next chapter.

Skill checkpointTarget
Concept theory & formulas understood100%
Topic practice sets attempted (4 topics)4/4
Best topic-test score— → 80%+
Chapter test score— → 80%+
Flashcards drilled to instant recall12 cards