SPI Mathematics · Mensuration 3D

Cuboids and Cubes

A cuboid has six rectangular faces in three equal pairs; its surface is the sum of their areas, its volume is length times breadth times height.

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Why this matters

Eleven questions in the bank, three of them from the 2022 paper: four EASY, six MODERATE and one HARD. They cover painting a box, sheet metal, bricks in a wall, earth from a pit and the longest rod in a room. The formulas are short; the marks go on units (inches, litres) and on which faces are counted.

Concept 1 of 3: Surface area of a cuboid and a cube

Unfold a box and you get six rectangles: top and bottom, front and back, two ends. Each pair is the same size, so you find three areas, add them and double.

Definition

Total surface of a cuboid: $2(lb + bh + hl)$ . For a cube of edge $a$ , all six faces are $a^2$ , so $6a^2$ .
Lateral surface (the four walls, no top or bottom): $2(l + b)h$ . For a cube, $4a^2$ .
Cubes joined in a row: each joint hides two faces. Two cubes of edge $a$ give $10a^2$ , three give $14a^2$ .
Change units before you square. If a rate is per square inch, turn each edge into inches first. The 2022 paper fixes 1 inch $= 2.5$ cm, so 1 square inch $= 6.25$ square cm.

Surface of a cuboid and a cube

S = 2(lb + bh + hl), \quad \text{walls} = 2(l + b)h, \quad S_{\text{cube}} = 6a^2, \quad \text{walls}_{\text{cube}} = 4a^2

l, b, hlength, breadth and height of the cuboid
aedge of the cube

Worked example

A closed box is 8 cm long, 5 cm wide and 3 cm high. Sheet costs ₹2 per square cm. Find the cost of the sheet for the box.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q55Moderate

Example 1 · Mensuration 3D · Cuboids and Cubes

Answer the next four questions based on the information given below A cube, with each being 10cm is painted in such a manner that red is opposite black. Green is between red and black. Blue is adjacent to white. Pink is adjacent to blue and red is at bottom. Rates of painting different colours per square inch are:- red ₹2/-, black, white and blue ₹3/-, green and pink ₹4/-.

If the complete cube is to be painted in white, the expenditure will be ₹____.

Total surface or four walls

Painting the sides only, or the walls of a room, leaves out the top and bottom:

2(l + b)h

, or

4a^2

for a cube. The whole outside is

6a^2

. Read which one is asked before you multiply.

Converting an area by the length factor

If 1 inch is 2.5 cm, then 1 square inch is

2.5^2 = 6.25

square cm, not 2.5. Convert each edge to inches first, then find the area.

Counting the hidden faces

Three separate cubes of edge

a

show

18a^2

. Joined in a row, two joints hide four faces, so the cuboid shows

14a^2

Drill 5 more on surface area of a cuboid and a cube

Concept 2 of 3: Volume of a cuboid, litres and how many blocks fit

Volume counts how many unit cubes fill the box: a layer of length times breadth, stacked height times. Most questions then turn that volume into litres, bricks or a rise in level.

Definition

Volume $= l \times b \times h$ , with all three lengths in the same unit. A cube: $a^3$ .
Litres: 1 litre $= 1000$ cubic cm, and 1 cubic m $= 1000$ litres.
How many blocks: divide the big volume by one block's volume. If mortar fills a share of a wall, the bricks fill only the rest.
Earth spread over land: rise in level $=$ volume of earth $\div$ area it is spread over. Leave the pit's own area out.

Volume and units

V = lbh, \quad V_{\text{cube}} = a^3, \quad 1\ \text{litre} = 1000\ \text{cm}^3, \quad 1\ \text{m}^3 = 1000\ \text{litres}, \quad \text{rise} = \frac{\text{volume}}{\text{area}}

Worked example

A tank is 2 m long, 1.5 m wide and 1 m deep. How many 10-litre buckets of water fill it?

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Mensuration 3D · Cuboids and Cubes

The dimensions of a field are 12 m

\times

10 m. A pit 5 m long, 4 m wide and 2 m deep is dug in one corner of the field and the earth removed has been evenly spread over the remaining area of the field. The level of the field is raised by

Mixing units inside one product

A tank 4 m by 150 cm by 1 m must be written as 400 cm by 150 cm by 100 cm (or all in metres) before multiplying. And 1 cubic m is 1000 litres, not 100.

Leaving the pit in the area

Earth dug from a pit is spread over the rest of the field, so subtract the pit's floor area from the field's area before dividing.

Forgetting the mortar

If one eighth of a wall is mortar, the bricks fill seven eighths of its volume. Take that share first, then divide by one brick's volume.

Dividing volumes when the blocks cannot fit

Cubes of edge 4 cm from a block 12 cm by 10 cm by 8 cm: the volumes divide to 15, but only

3 \times 2 \times 2 = 12

whole cubes can be cut, because 10 is not a multiple of 4. Check each edge when the question says cut.

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Concept 3 of 3: The longest rod: the diagonal of a cuboid

The longest straight rod in a room runs from one bottom corner to the opposite top corner. Pythagoras on the floor gives the floor diagonal; Pythagoras once more with the height gives the rod.

Definition

Floor diagonal $= \sqrt{l^2 + b^2}$ .
Space diagonal $= \sqrt{l^2 + b^2 + h^2}$ . This is the longest rod.
For a cube of edge $a$ : $\sqrt{3a^2} = a\sqrt{3}$ .
Simplify the root by taking out the largest square: $\sqrt{72} = 6\sqrt{2}$ .

Diagonal of a cuboid

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

Worked example

Find the length of the longest pencil that fits in a box 12 cm by 4 cm by 3 cm.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Mensuration 3D · Cuboids and Cubes

Find out the length of the longest rod which can be kept in the hall of 10 m length, 6 m breadth and 4 m height?

Stopping at the floor diagonal

\sqrt{l^2 + b^2}

is the longest rod that lies flat on the floor. The longest rod in the room also uses the height:

\sqrt{l^2 + b^2 + h^2}

Leaving the root unsimplified

Options print roots in simplest form:

\sqrt{200} = 10\sqrt{2}

. Take out the largest square factor before matching.

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)

Surface area of a cuboid and a cube
Surface of a cuboid and a cube
$S = 2(lb + bh + hl), \quad \text{walls} = 2(l + b)h, \quad S_{\text{cube}} = 6a^2, \quad \text{walls}_{\text{cube}} = 4a^2$
Volume of a cuboid, litres and how many blocks fit
Volume and units
$V = lbh, \quad V_{\text{cube}} = a^3, \quad 1\ \text{litre} = 1000\ \text{cm}^3, \quad 1\ \text{m}^3 = 1000\ \text{litres}, \quad \text{rise} = \frac{\text{volume}}{\text{area}}$
The longest rod: the diagonal of a cuboid
Diagonal of a cuboid
$d = \sqrt{l^2 + b^2 + h^2}, \qquad d_{\text{cube}} = a\sqrt{3}$

Watch out for (9)

Total surface or four walls
→ Surface area of a cuboid and a cube
Converting an area by the length factor
→ Surface area of a cuboid and a cube
Counting the hidden faces
→ Surface area of a cuboid and a cube
Mixing units inside one product
→ Volume of a cuboid, litres and how many blocks fit
Leaving the pit in the area
→ Volume of a cuboid, litres and how many blocks fit
Forgetting the mortar
→ Volume of a cuboid, litres and how many blocks fit
Dividing volumes when the blocks cannot fit
→ Volume of a cuboid, litres and how many blocks fit
Stopping at the floor diagonal
→ The longest rod: the diagonal of a cuboid
Leaving the root unsimplified
→ The longest rod: the diagonal of a cuboid

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