CDS Mathematics · Mensuration 3D

Cubes & Cuboids

Volume and surface area of cubes and cuboids: the walls of a room, blocks joined or cut, and open boxes with walls of some thickness.

All Mensuration 3D notes CDS Mathematics strategy

Why this matters

A steady, mostly MODERATE page. The formulas are the easiest in the chapter; the marks are lost on reading — which faces are counted, whether the top is open, and how many faces disappear when blocks are joined.

Concept 1 of 3: Walls, edges and faces

Before any formula, count what the question counts. Four walls leave out the floor and the ceiling. A skeleton or a taped cube uses the twelve edges. Paint and tiles cover faces.

Definition

For an $l \times b \times h$ cuboid:

Volume $lbh$ ; total surface $2(lb + bh + hl)$ .
Four walls only $= 2(l + b)h$ , the perimeter of the floor times the height.
A cube has $6$ faces and $12$ edges; its surface is $6a^2$ .
Buying paint or tiles: divide, then round up to whole cans or packets.

Four walls of a room

\text{walls} = 2(l + b)\,h

Worked example

A room is

8

m long,

5

m wide and

3

m high. One litre of paint covers

16

^2

, and paint is sold in whole litres. How many litres are needed for the four walls?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — Elementary Mathematics · Q41Moderate

Example 1 · Mensuration 3D · Cubes and Cuboids

The area of four walls of a room is

120\ \text{m}^2

. The length of the room is twice its breadth. If the height of the room is 4 m, what is area of the floor ?

Round the cans up, never down

12.5

litres of paint means

13

one-litre cans. The rounded-down value is always one of the options.

Drill 6 more on walls, edges and faces

Concept 2 of 3: Joining and cutting blocks

Joining blocks keeps the volume but hides the faces that touch. Cutting a block into cubes keeps the volume too; the fewest cubes come from the largest edge that divides every dimension.

Definition

$n$ cubes of edge $a$ in a row: a cuboid $na\times a\times a$ with surface $(4n + 2)a^2$ . Each join hides two faces.
Fewest equal cubes from an $l \times b \times h$ block: edge $= \gcd(l, b, h)$ .
Bricks in a wall: $\dfrac{\text{wall volume}}{\text{brick volume}}$ , both in the same unit.
Edges that are consecutive, prime or natural numbers: factorise the volume.

n cubes in a row

\text{surface} = (4n + 2)\,a^2

Worked example

Four cubes of edge

5

cm are joined end to end. Find the surface area of the cuboid.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2019 · CDS (I) 2019 — Elementary Mathematics · Q70Moderate

Example 2 · Mensuration 3D · Cubes and Cuboids

Six cubes, each with 12 cm edge are joined end to end. What is the surface area of resulting cuboid ?

Drill 7 more on joining and cutting blocks

Concept 3 of 3: Open boxes and wall thickness

A box made by cutting squares from a sheet's corners has the cut size as its height and loses twice the cut from each side. A box with thick walls is the outer block minus the inner block; the bottom counts once, the sides twice.

Definition

Sheet $L\times B$ , corners of side $x$ cut out: box $(L - 2x)\times(B - 2x)\times x$ . Test the options, since the equation is a cubic.
Closed box of wall thickness $t$ , external $L\times B\times H$ : internal $(L - 2t)(B - 2t)(H - 2t)$ .
Open box with sides $s$ thick and bottom $t$ thick: external height $=$ internal $+ t$ only.
Material $=$ external volume $-$ internal volume.

Box from a sheet

V = x(L - 2x)(B - 2x)

Worked example

Squares of side

3

cm are cut from the corners of a

20\times 16

cm sheet and the sides folded up. Find the volume of the box.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2022 · CDS (II) 2022 — Elementary Mathematics · Q91Moderate

Example 3 · Mensuration 3D · Cubes and Cuboids

A closed box is built of wood of uniform thickness. Its external dimensions are 12 cm, 10 cm and 8 cm. If the inner surface area is 376 square cm, then what is the thickness of the wood ?

The open top has no wall

An open box adds thickness to the height only once, at the bottom. Adding it twice, as for a closed box, overstates the material.

Drill 4 more on open boxes and wall thickness

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 (3)

Walls, edges and faces
Four walls of a room
$\text{walls} = 2(l + b)\,h$
Joining and cutting blocks
n cubes in a row
$\text{surface} = (4n + 2)\,a^2$
Open boxes and wall thickness
Box from a sheet
$V = x(L - 2x)(B - 2x)$

Watch out for (2)

Round the cans up, never down
→ Walls, edges and faces
The open top has no wall
→ Open boxes and wall thickness

Test yourself on Mensuration 3D

20 past CDS questions from this chapter, timed at 24 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 20 questions