Mensuration (3D)

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

Overview Learn (5)Formulas (21)Examples (4)Practice (64)

Cube and cuboid

⏱ 4 min read🧩 5 question types🎯 14 practice Q

The idea in one minute

A cuboid has three dimensions l, b, h; a cube makes them equal. Volume is the product of the three; surface area adds the six faces. Capacity questions are volume questions with a litres conversion at the end.

The master table

Quantity	Cuboid $l\times b\times h$	Cube, edge $a$
Volume	$lbh$	$a^3$
Lateral surface (4 walls)	$2h(l+b)$	$4a^2$
Total surface (6 faces)	$2(lb+bh+hl)$	$6a^2$
Diagonal	$\sqrt{l^2+b^2+h^2}$	$a\sqrt3$

Cuboid $12\times10\times8$ : volume $960$ , walls $2\times8\times22=352$ , total surface $2\times(120+80+96)=592$ .

Reverse works too: volume $960$ with two dimensions $12$ and $10$ leaves the third as $\dfrac{960}{120}=8$ . Attach square units to areas and cubic units to volumes, always.

Rule: Total surface is the walls plus top and bottom. Lateral surface alone skips the two $l\times b$ faces.

Cube shortcuts

Edge $6$ : volume $216$ , lateral $144$ , total $216$ , diagonal $6\sqrt3$ . The diagonal formula comes from using the cuboid diagonal with all three sides equal. A cube of volume $216$ has edge $6$ ; volume $125$ gives edge $5$ .

Doubling every edge multiplies area by $4$ and volume by $8$ . Edge $5\to10$ : total surface $150\to600$ , volume $125\to1000$ .

Tip: Lengths take $k$ , areas $k^2$ , volumes $k^3$ . Count what kind of quantity is asked before multiplying anything.

Capacity and cost

One cubic metre holds $1000$ litres, one litre is $1000$ cubic centimetres. A tank $1.5\times1\times0.8$ m has volume $1.2$ m $^3$ , which is $1200$ litres.

Cost questions multiply volume by a rate: digging at Rs $20$ per m $^3$ costs $20\times1.2=$ Rs $24$ for that tank. Match the rate's units to the volume's units before multiplying.

Watch: Convert units exactly once, at the end. Mixing cm and m midway is where capacity questions die.

Reverse questions

Given the volume or surface, peel back to the edge, then forward to the asked quantity.

Volume $343$ cu cm means edge $7$ (since $7^3=343$ ); total surface is then $6\times49=294$ sq cm. Two steps, no algebra.

Remember: Learn the cubes up to $12$ : $1,8,27,64,125,216,343,512,729,1000,1331,1728$ . Reverse questions then need no cube roots.

Painted cubes

An $n\times n\times n$ cube painted outside, cut into unit cubes:

$3$ faces painted: the $8$ corners
$2$ faces: the edges, $12(n-2)$
$1$ face: face centres, $6(n-2)^2$
$0$ faces: inside, $(n-2)^3$

A $4$ cm cube cut into $1$ cm cubes: $8$ , $24$ , $24$ , $8$ . The four counts always add to $n^3=64$ . A $3$ cm cube gives $8$ , $12$ , $6$ , $1$ .

Tip: Only corner cubes get three faces, always exactly $8$ . Start there, then fill the edge and face counts. Slice counts never change when the whole cube is dipped instead of brushed.

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Volume and surface area of cuboid/cube

How to spot it:

Three dimensions given, volume or a surface asked.

Method

Volume: multiply all three.
Lateral: $2h(l+b)$ .
Total: $2(lb+bh+hl)$ , or $6a^2$ for a cube.
Attach square or cubic units.

Why it works:

Both formulas are one substitution each once the dimensions are named.

Try this

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

Show solution

$2\times(12\times10+10\times8+12\times8)$ .
$=2\times296=592$ sq cm.

Answer

592 sq cm

Type 2common3 practice Q

Cube specifics: diagonal, lateral area, edges

How to spot it:

A cube with one length given; a diagonal or surface asked.

Method

Write the cube formulas in terms of the edge $a$ .
Diagonal $=a\sqrt3$ .
Lateral $=4a^2$ , total $=6a^2$ .

Why it works:

One variable drives every cube quantity, so a single given length answers everything.

Try this

The length of the diagonal of a cube of side 6 cm is:

Show solution

$d=a\sqrt3$ .
$=6\sqrt3$ cm.

Answer

6*sqrt(3) cm

Type 3very common2 practice Q

Tank capacity in litres / volume cost

How to spot it:

A tank, pit or room with dimensions; litres or rupees asked.

Method

Compute the volume in cubic metres.
Capacity: multiply by $1000$ .
Cost: multiply by the rate per cubic metre.

Why it works:

Capacity and cost are both volume times a constant, so the geometry happens once.

Try this

A tank is 1.5 m long, 1 m wide and 0.8 m deep. Its capacity in litres is:

Show solution

$V=1.2$ m $^3$ .
$1.2\times1000=1200$ litres.

Answer

1200 litres

Type 4common2 practice Q

Reverse: from volume or area back to a side

How to spot it:

A volume or surface area given, an edge or another quantity asked.

Method

Match the volume to a known cube ( $7^3=343$ ).
Recover the edge.
Substitute into the asked formula.

Why it works:

Reverse questions are look-ups when the cube table is memorised, algebra when it is not.

Try this

The volume of a cube is 343 cu cm. Its total surface area is:

Show solution

$343=7^3$ , edge $=7$ cm.
TSA $=6\times49=294$ sq cm.

Answer

294 sq cm

Type 5common

Painted cube cut into unit cubes

How to spot it:

A big painted cube sliced into equal small cubes; a count asked.

Method

Find $n$ : big edge over small edge.
Three painted faces: $8$ corners.
Two: $12(n-2)$ ; one: $6(n-2)^2$ ; none: $(n-2)^3$ .

Why it works:

Position, not luck, decides how many painted faces a small cube carries; the counts are pure formulas.

Try this

A cube of 4 cm edge is painted on all faces and cut into 1 cm cubes. How many small cubes have exactly two painted faces?

Show solution

$n=\dfrac{4}{1}=4$ .
Two faces: $12(n-2)=12\times2=24$ .

Answer

Formula sheet

Cuboid

V=lbh,\quad \text{LSA}=2h(l+b),\quad \text{TSA}=2(lb+bh+hl)

l, b, h are the three dimensions.

Cube

V=a^3,\quad \text{LSA}=4a^2,\quad \text{TSA}=6a^2

a = edge.

Diagonals

d=\sqrt{l^2+b^2+h^2},\quad d_{\text{cube}}=a\sqrt3

Corner to opposite corner.

Capacity

1\text{ m}^3=1000\text{ L},\quad 1\text{ L}=1000\text{ cm}^3

Convert once, at the end.

Painted cube counts

8,\ 12(n-2),\ 6(n-2)^2,\ (n-2)^3

3, 2, 1, 0 painted faces for n by n by n.

Shortcuts that save time

⚡ Cubes table, not cube roots

Know the cubes to 12 by heart. Reverse questions become look-ups: 343 is 7 cubed.

Example

The volume of a cube is 343 cu cm. Its total surface area is:

Show solution

$343=7^3$ , so the edge is $7$ cm.
TSA $=6\times7^2$ .
$=294$ sq cm.

Answer

294 sq cm

⚡ Capacity: metres to litres

Compute the volume in cubic metres, then multiply by 1000 for litres. Never by 100.

Example

A tank is 1.5 m long, 1 m wide and 0.8 m deep. Its capacity in litres is:

Show solution

$V=1.5\times1\times0.8=1.2$ m $^3$ .
$1.2\times1000$ .
$=1200$ litres.

Answer

1200 litres

⚡ Edge scaling in powers

Edge times k: surface times k squared, volume times k cubed. Doubling is 4 and 8.

Example

Each edge of a cube of edge 5 cm is doubled. The total surface area becomes:

Show solution

Old TSA $=6\times25=150$ .
Area scales by $2^2=4$ .
$150\times4=600$ sq cm.

Answer

600 sq cm

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Using $6lb$ as the cuboid total surface.

It is $2(lb+bh+hl)$ . For $12\times10\times8$ that is 592, not 720.

Mistake 02

Forgetting the two $lb$ faces in total surface.

Total = lateral + top + bottom. Lateral $352$ plus $2\times120=240$ gives 592.

Mistake 03

Multiplying cubic metres by 100 for litres.

One cubic metre is $1000$ litres: $1.2$ m $^3$ is 1200 L.

Mistake 04

Doubling the edge and doubling the volume.

Volume takes the cube of the scale: $2^3=8$ times, so $125\to1000$ .

Mistake 05

Painted-cube corner counts other than 8.

Only the 8 corners carry three painted faces, whatever $n$ is.

Quick revision

Read this the night before the exam.

Cuboid: $V=lbh$ , walls $2h(l+b)$ , total $2(lb+bh+hl)$ .
Cube: $V=a^3$ , walls $4a^2$ , total $6a^2$ , diagonal $a\sqrt3$ .
$1$ m $^3=1000$ L and $1$ L $=1000$ cm $^3$ ; convert once at the end.
Reverse questions: recall the cube from the table ( $343\to7$ ), then move forward.
Scale $k$ : areas $\times k^2$ , volumes $\times k^3$ .
Painted $n$ -cube: $8$ corners, $12(n-2)$ edges, $6(n-2)^2$ faces, $(n-2)^3$ inside.

Practice: 14 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 10 questions

Suggested time 5 min · wrong answers go to your mistake notebook automatically.

Continue learning