Mensuration (3D)

medium importance~2 Q in Tier 121 formulas⚡ 15 shortcuts5 subtopics

Overview Learn (5)Formulas (21)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.

Box (cuboid): volume and surface

very common

Length, breadth and height of a box are given. Volume or surface area is asked.

Volume = length × breadth × height.

Surface = 2 × (sum of the three different face areas).

Example

Find the total surface area of a cuboid 12 × 10 × 8 cm.

Faces: 12 × 10 = 120, 10 × 8 = 80, 12 × 8 = 96

Sum = 296

Surface = 2 × 296 = 592 sq cm

Learn this in “Cube and cuboid” →

Cube: body diagonal

common

A cube's surface area or volume is given. Its diagonal is asked.

Find the edge first.

Body diagonal = edge × √3.

Example

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

A cube has 6 faces, so one face = 294 ÷ 6 = 49

Edge = √49 = 7

Diagonal = 7 × √3 = 7√3 cm

Learn this in “Cube and cuboid” →

Tank capacity in litres

very common

"How many litres does the tank hold?" or digging cost.

Multiply length × breadth × height in metres.

1 cubic metre = 1000 litres.

Example

A tank is 1.5 m × 1 m × 0.8 m. How many litres does it hold?

Volume = 1.5 × 1 × 0.8 = 1.2 cubic m

1 cubic m = 1000 litres

1.2 × 1000 = 1200 litres

Learn this in “Cube and cuboid” →

Cylinder: volume and surface

very common

Radius and height of a cylinder. Volume or surface area is asked.

Volume = π × r × r × height.

Curved surface = 2 × π × r × height.

Example

Find the volume of a cylinder with radius 7 cm and height 10 cm.

Base area = 22/7 × 7 × 7 = 154

Volume = base area × height

= 154 × 10 = 1540 cubic cm

Learn this in “Cylinder” →

Cylinder: changing the size

common

"Radius is doubled and height is halved. What happens to the volume?"

Try a simple radius and height.

Compare old and new volume.

Example

The radius of a cylinder is doubled and its height halved. Its volume becomes how many times?

Take r = 1, h = 2: r × r × h = 2

New r = 2, h = 1: 2 × 2 × 1 = 4

4 ÷ 2 = 2 times

Learn this in “Cylinder” →

Hollow pipe

common

A pipe with an outer and an inner radius. Volume of metal is asked.

Metal = outer circle area − inner circle area.

Then multiply by the length.

Example

An iron pipe 14 cm long has outer radius 5 cm and inner radius 3 cm. Find the volume of iron.

Ring area = 22/7 × (5 × 5 − 3 × 3) = 22/7 × 16

Volume = ring area × 14

= 22 × 16 × 2 = 704 cubic cm

Learn this in “Cylinder” →

Cone: slant height and surface

very common

Radius and height of a cone. Surface area is asked.

Slant height = √(r² + h²). Watch for 7, 24, 25.

Curved surface = π × r × slant height.

Example

A cone has r = 7 cm and h = 24 cm. Find its curved and total surface area.

Slant = √(49 + 576) = 25

Curved = 22/7 × 7 × 25 = 550

Base = 22/7 × 49 = 154

Total = 550 + 154 = 704 sq cm

Learn this in “Cone” →

Melting and recasting

very common

"A solid is melted and made into another shape."

Volume stays the same.

Write both volumes and set them equal.

Example

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

Cone = ⅓ × π × 6 × 6 × 24 = 288π

Sphere = 4/3 × π × R³ = 288π

R³ = 288 × 3 ÷ 4 = 216

R = 6 cm

Learn this in “Cone” →

Sphere and hemisphere

very common

A ball, bowl or dome. Surface area or volume is asked.

Sphere surface = 4 × π × r × r.

Sphere volume = 4/3 × π × r × r × r.

Example

Find the volume of a sphere of radius 21 cm (π = 22/7).

21 × 21 × 21 = 9261

4/3 × 22/7 × 9261 = 4 × 22 × 1323 ÷ 3

= 4 × 22 × 441 = 38808 cubic cm

Learn this in “Sphere and hemisphere” →

Prism, pyramid and bucket

common

A prism, a pyramid or a bucket-shaped solid (frustum). Volume is asked.

Prism: base area × height. Pyramid: one-third of that.

Bucket: ⅓ × π × height × (R×R + r×r + R×r).

Example

A bucket has radii 5 cm and 3 cm and height 6 cm. Find its capacity.

R × R + r × r + R × r = 25 + 9 + 15 = 49

⅓ × 22/7 × 6 × 49

= 22/7 × 2 × 49 = 308 cubic cm

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

Your next step

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