SSC CGL Maths · Mensuration 3D

Cubes and Cuboids

A cuboid holds length times breadth times height, and its surface is six rectangles in three equal pairs. A cube is the cuboid whose three edges are equal.

All Mensuration 3D notes

Why this matters

Thirty-one questions in the bank, twenty-seven EASY and four MODERATE, fourteen of them from the 2023 shifts. They test four things: volume and surface from the edges (or the edges back from them), the walls of a room and the capacity of a tank, cutting and joining blocks, and the longest diagonal. Every later page reuses the habit started here: write the formula, put the numbers in, and simplify before you multiply.

Concept 1 of 4: Volume and surface area of a cuboid and a cube

Volume counts the unit cubes that fill the box: one layer holds length times breadth of them, and there are as many layers as the height. Surface area is the paper you need to wrap the box: six rectangles, two of each size.

Definition

Volume of a cuboid $= lbh$ . For a cube of edge $a$ it is $a^3$ .
Total surface area of a cuboid $= 2(lb + bh + hl)$ . For a cube it is $6a^2$ : six equal square faces.
Going back: from the surface of a cube, divide by 6 and take the square root; from its volume, take the cube root.
Fast cube root of a perfect cube: the last digit of the cube fixes the last digit of the root (a cube ending in 2 comes from a number ending in 8, 3 from 7, 7 from 3, 8 from 2; every other digit gives itself). Then drop the last three digits and see which two cubes the rest lies between. For 13824: last digit 4 gives 4; 13 lies between $2^3 = 8$ and $3^3 = 27$ , so the root is 24. This works only when the number is a perfect cube.
Edges in a ratio $a : b : c$ : write them as $ak$ , $bk$ , $ck$ . The volume is $abc\,k^3$ , so find $k^3$ first.

Cuboid and cube

V = lbh \qquad S = 2(lb + bh + hl) \qquad V_{\text{cube}} = a^3 \qquad S_{\text{cube}} = 6a^2

$l, b, h$ length, breadth and height of the cuboid
$a$ edge of the cube

Worked example

A cuboid is 9 cm long, 5 cm wide and 4 cm high. Find its volume and its total surface area.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 19 Sep 2025, Shift 1 · Q62Easy

Example 1 · Mensuration 3D · Cubes and Cuboids

The sides of a rectangular prism are in the ratio 4:5:6. If the volume of the prism is

960\ \text{cm}^3

what is the length of its longest side?

Leaving out the 2 in the surface formula

The total surface of a cuboid is

2(lb + bh + hl)

. For a cuboid 2 by 3 by 4 the bracket is 26 but the surface is 52; the half value is often a printed option.

Edge from the perimeter of one face

A face of a cube is a square, so its perimeter is

4a

. A face perimeter of 36 cm gives an edge of 9 cm. The total of all twelve edges,

12a

, is a different measure.

Drill 14 more on volume and surface area of a cuboid and a cube

Concept 2 of 4: Walls of a room and the capacity of a tank

Unfold the four walls of a room and they make one long strip: its length is the distance round the floor,

2(l + b)

, and its height is the room's height. The floor and the ceiling are separate rectangles that you add only when the question names them.

Definition

Four walls $= 2(l + b)h$ : walk once round the floor, and multiply that distance by the height.
Add the ceiling $lb$ or the floor $lb$ only when the question asks for them. A box open at the top has its floor and four walls: $lb + 2(l + b)h$ .
Cost $=$ area $\times$ rate. A rate such as ₹40 per $20\ \text{m}^2$ is first turned into ₹2 per $\text{m}^2$ .
Capacity: $1\ \text{m}^3 = 1000$ litres $= 1$ kilolitre, and $1000\ \text{cm}^3 = 1$ litre. Put every length in one unit before you multiply.
Wall thickness: a closed box of outer size $l \times b \times h$ with walls $t$ thick has inner size $(l - 2t)(b - 2t)(h - 2t)$ . The wood used is the outer volume minus the inner volume.
Whole items: to cover or fill something, round up (13.1 cans of paint means buying 14). To count how many whole items fit into a space, round down.

Four walls and capacity

A_{\text{walls}} = 2(l + b)h \qquad 1\ \text{m}^3 = 1000\ \text{L} \qquad 1\ \text{L} = 1000\ \text{cm}^3

Worked example

A room is 12 m long, 7 m wide and 4 m high. Find the cost of painting its four walls and its ceiling at ₹15 per

\text{m}^2

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 23 Sep 2025, Shift 1 · Q55Moderate

Example 2 · Mensuration 3D · Cubes and Cuboids

A large wooden box, open at the top, needs to be lined with fabric on its inner floor and inner four side walls. The inner dimensions of the box are 2.5 m length, 1.8 m wide, and 1.2 m high. If the fabric comes in square pieces of side 50 cm, and

15\%

extra fabric is needed for cuts and overlaps, how many approximate fabric pieces should be purchased?

Taking the thickness off once

A wall 1 cm thick sits at both ends of every edge, so each inner edge is 2 cm shorter than the outer edge, not 1 cm shorter.

Mixing metres and centimetres

A depth of 80 cm is 0.8 m. Change every length to one unit before multiplying, then use

1\ \text{m}^3 = 1000

litres or

1000\ \text{cm}^3 = 1

litre.

Adding the floor to the walls by habit

Whitewashing walls and ceiling leaves out the floor; painting only the walls leaves out both. Add

lb

only for each surface the question names.

Drill 7 more on walls of a room and the capacity of a tank

Concept 3 of 4: Cutting a cuboid into cubes and joining cubes

A big block cut into equal small blocks keeps its volume, so the count is the big volume over the small one. The surface does change: every cut makes two new faces and every join hides two.

Definition

Largest equal cubes from an $l \times b \times h$ block with no waste: edge $a =$ HCF of $l$ , $b$ , $h$ ; count $= \frac{l}{a} \times \frac{b}{a} \times \frac{h}{a}$ .
Bricks in a wall: count $= \frac{\text{wall volume}}{\text{one brick's volume}}$ , both in the same unit.
Cubes in a row: $n$ cubes of edge $a$ joined end to end make an $na \times a \times a$ cuboid. Its surface is $(4n + 2)a^2$ , because each join hides two faces.
Straight cuts: to split the three edges into $p$ , $q$ and $r$ parts you need $(p - 1) + (q - 1) + (r - 1)$ cuts and get $pqr$ pieces. For the fewest cuts, share the parts as evenly as you can among the three directions.

Cutting and joining

\text{Count} = \frac{V_{\text{big}}}{V_{\text{small}}} \qquad S_{n \text{ cubes in a row}} = (4n + 2)a^2

Worked example

A block of wood measures 36 cm by 24 cm by 60 cm. It is cut into the largest identical cubes possible with no waste. How many cubes are there?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 17 Sep 2025, Shift 3 · Q62Easy

Example 3 · Mensuration 3D · Cubes and Cuboids

A large cubical cake needs to be cut into 64 smaller, identical cubical pieces. What is the minimum number of straight cuts required?

Adding the surfaces of joined cubes

Three separate cubes show 18 faces, but joined in a row they show only 14: the two faces at each join are hidden. The surface of

n

cubes in a row is

(4n + 2)a^2

, not

6na^2

One cut for every part

Dividing one edge into 4 parts needs 3 cuts, not 4: the last part needs no cut of its own.

Drill 4 more on cutting a cuboid into cubes and joining cubes

Concept 4 of 4: The longest diagonal of a cuboid and a cube

The longest straight rod that fits in a box runs from one bottom corner to the opposite top corner. Use Pythagoras twice: across the floor, then up.

Definition

Body diagonal of a cuboid $= \sqrt{l^2 + b^2 + h^2}$ .
For a cube of edge $a$ this is $a\sqrt{3}$ . The diagonal of one face is only $a\sqrt{2}$ .
The longest stick or rod that fits inside a box lies along the body diagonal.

Body diagonal

d = \sqrt{l^2 + b^2 + h^2} \qquad d_{\text{cube}} = a\sqrt{3} \qquad d_{\text{face}} = a\sqrt{2}

Worked example

Find the length of the longest rod that fits inside a box 12 cm long, 4 cm wide and 3 cm high.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SSC CGL · 2024 · Tier I, 17 Sep 2024, Shift 3 · Q56Easy

Example 4 · Mensuration 3D · Cubes and Cuboids

Find the length of the longest diagonal (in cm) of a cuboidal box of dimensions 1.2 cm

\times

1.3 cm

\times

1.5 cm (correct to two decimal places).

Using the face diagonal for the longest rod

a\sqrt{2}

runs across one face. The rod through the inside of a cube is longer:

a\sqrt{3}

Drill 2 more on the longest diagonal of a cuboid and a cube

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (4)

Volume and surface area of a cuboid and a cube
Cuboid and cube
$V = lbh \qquad S = 2(lb + bh + hl) \qquad V_{\text{cube}} = a^3 \qquad S_{\text{cube}} = 6a^2$
Walls of a room and the capacity of a tank
Four walls and capacity
$A_{\text{walls}} = 2(l + b)h \qquad 1\ \text{m}^3 = 1000\ \text{L} \qquad 1\ \text{L} = 1000\ \text{cm}^3$
Cutting a cuboid into cubes and joining cubes
Cutting and joining
$\text{Count} = \frac{V_{\text{big}}}{V_{\text{small}}} \qquad S_{n \text{ cubes in a row}} = (4n + 2)a^2$
The longest diagonal of a cuboid and a cube
Body diagonal
$d = \sqrt{l^2 + b^2 + h^2} \qquad d_{\text{cube}} = a\sqrt{3} \qquad d_{\text{face}} = a\sqrt{2}$

Watch out for (8)

Leaving out the 2 in the surface formula
→ Volume and surface area of a cuboid and a cube
Edge from the perimeter of one face
→ Volume and surface area of a cuboid and a cube
Taking the thickness off once
→ Walls of a room and the capacity of a tank
Mixing metres and centimetres
→ Walls of a room and the capacity of a tank
Adding the floor to the walls by habit
→ Walls of a room and the capacity of a tank
Adding the surfaces of joined cubes
→ Cutting a cuboid into cubes and joining cubes
One cut for every part
→ Cutting a cuboid into cubes and joining cubes
Using the face diagonal for the longest rod
→ The longest diagonal of a cuboid and a cube

Test yourself on Mensuration 3D

20 past SSC CGL questions from this chapter, timed at 12 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Mensuration 3D chapter test Drill all 31 questions