Mensuration (3D)

medium importance~2 Q in Tier 120 formulas⚡ 10 shortcuts5 subtopics

Overview Learn (5)Formulas (20)Examples (4)Practice (64)

Volumes and surface areas of cubes, cuboids, cylinders, cones, spheres, hemispheres, prisms and pyramids, plus melting-recasting, capacities and painted-cube counts. About 1-3 questions per shift; every question reduces to one formula plus a unit conversion, so memorising the formula sheet is nearly the whole job.

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Learn — 5 subtopics

One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.

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Formula sheet

Every formula on one printable page, grouped by subtopic.

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Solved examples

4 exam-level questions worked step by step.

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Practice & test

64 questions — untimed practice or a timed test with analysis.

Track record in the exam

avg 1.0 Q / shift2024: 1 Q2025: 1 Q

Questions per shift in recent SSC CGL papers.

Test difficulty mix (64 questions)

17 easy37 medium10 hard

Question patterns exams keep repeating

Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.

Cuboid/cube: volume, total/lateral surface area

very common

Spot it:

Dimensions of a box given; volume, total surface, or the four walls of a room asked.

How to solve: $V = lbh = a^3$ ; TSA $= 2(lb+bh+lh) = 6a^2$ ; four walls $= 2h(l+b)$ . Read which measure the wording prices.

Example: Find the total surface area of a cuboid of dimensions $12 \times 10 \times 8$ cm.

TSA $= 2(120 + 80 + 96) = 592$ sq cm.

Learn this in “Cube and cuboid” →

Cube diagonal and edge-sum

common

Spot it:

A cube's body/face diagonal or the total edge length asked; or volume/area given and the diagonal asked.

How to solve: Body diagonal $a\sqrt3$ , face diagonal $a\sqrt2$ , edges $12a$ . Reverse from $V$ or TSA to $a$ first.

Example: The total surface area of a cube is $294$ sq cm. Find its body diagonal.

$a = 7$ ; diagonal $= 7\sqrt3$ cm.

Learn this in “Cube and cuboid” →

Tank/pit capacity in litres, digging cost

very common

Spot it:

A tank/pit/room's capacity asked in litres, or a cost at Rs x per cu m / per sq m.

How to solve: $1\ \text{m}^3 = 1000$ L; volume for capacity/digging, area for painting. Convert BEFORE applying rates.

Example: A tank $1.5$ m $\times$ $1$ m $\times$ $0.8$ m holds how many litres?

$1.2\ \text{m}^3 = 1200$ litres.

Learn this in “Cube and cuboid” →

Cylinder: volume / CSA / TSA

very common

Spot it:

Radius and height given (diameter must be halved); volume, curved or total surface asked.

How to solve: $V = \pi r^2 h$ , CSA $= 2\pi rh$ , TSA $= 2\pi r(h+r)$ with $\pi = 22/7$ . Capacity → $V$ , label → CSA, closed drum → TSA.

Example: Find the volume of a cylinder of radius $7$ cm and height $10$ cm.

$V = \frac{22}{7} \times 49 \times 10 = 1540$ cu cm.

Learn this in “Cylinder” →

Cylinder: reverse dimension or multiplier change

common

Spot it:

Volume/CSA given with a dimension asked — or 'radius doubled, height halved' style volume changes.

How to solve: $h = \frac{V}{\pi r^2}$ , $r = \frac{\text{CSA}}{2\pi h}$ ; multipliers: $r \times k$ contributes $k^2$ , $h \times m$ contributes $m$ .

Example: The radius of a cylinder is doubled and its height halved. Its volume becomes:

$4 \times \frac12 = 2$ times.

Learn this in “Cylinder” →

Hollow pipe, well capacity, water flow

common

Spot it:

A pipe with outer/inner radii (metal volume), a well (litres), or water flowing through a pipe.

How to solve: Pipe metal $= \pi(R^2 - r^2)L$ ; flow per second $= \pi r^2 \times$ speed; well litres $= 1000 \times \text{m}^3$ .

Example: An iron pipe of length $14$ cm has outer radius $5$ cm and inner radius $3$ cm. Find the volume of iron.

$\frac{22}{7} \times 16 \times 14 = 704$ cu cm.

Learn this in “Cylinder” →

Cone: slant triplet with CSA/TSA/volume

very common

Spot it:

Radius and height given; curved/total surface or volume asked — the $(r, h, l)$ triplet makes $l$ instant.

How to solve: $l = \sqrt{r^2+h^2}$ (7-24-25 family), CSA $= \pi r l$ , TSA $= \pi r(l+r)$ , $V = \frac13\pi r^2 h$ .

Example: A cone has $r = 7$ cm and $h = 24$ cm. Find its CSA and TSA.

$l = 25$ : CSA $= 550$ , TSA $= 550 + 154 = 704$ sq cm.

Learn this in “Cone” →

Recasting: cone/sphere/cylinder into each other

very common

Spot it:

'Melted and recast' — a solid's material reshaped; the target dimension asked.

How to solve: Equate volumes. Cone→sphere: $R^3 = \frac{r^2 h}{4}$ . Into $n$ similar solids: lengths divide by $\sqrt[3]{n}$ .

Example: A metallic cone of radius $6$ cm and height $24$ cm is melted into a sphere. Find the sphere's radius.

$R^3 = \frac{36 \times 24}{4} = 216 \Rightarrow R = 6$ cm.

Learn this in “Cone” →

Sphere/hemisphere: surface and volume

very common

Spot it:

A sphere or hemisphere (dome/bowl) with radius given or implied; SA, TSA, or volume asked.

How to solve: $SA = 4\pi r^2$ , $V = \frac43 \pi r^3$ ; hemisphere CSA $2\pi r^2$ , TSA $3\pi r^2$ , $V \frac23 \pi r^3$ . Radius $\times k$ : SA $k^2$ , V $k^3$ .

Example: Find the volume of a sphere of radius $21$ cm (take $\pi = 22/7$ ).

$\frac43 \times \frac{22}{7} \times 9261 = 38808$ cu cm.

Learn this in “Sphere and hemisphere” →

Prism, pyramid, frustum volumes

common

Spot it:

A constant cross-section solid (prism), a pointed solid (pyramid), or a bucket-shaped frustum with volumes asked.

How to solve: Prism $V = A \times h$ ; pyramid $\frac13 A h$ ; frustum $\frac{\pi h}{3}(R^2 + r^2 + Rr)$ . Cube melts: $\sqrt[3]{\sum a_i^3}$ .

Example: A bucket is a frustum with radii $5$ cm, $3$ cm and height $6$ cm. Find its capacity.

$\frac{22}{7} \times 2 \times (25+9+15) = 308$ cu cm.

Learn this in “Prisms, pyramids and painted cubes” →

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