MathematicsClass 8ICSE

The Longest Stick That Fits Inside a Box Is Its Diagonal

Learn the lateral and total surface area and volume of cuboids and cubes, find the space diagonal, convert between cubic units and litres, and solve painting and tank problems.

What is the longest rod that will fit inside a box?

Its space diagonal — the line from one corner to the corner furthest away, running right through the middle.

For a box

6

cm long,

3

cm wide and

2

cm deep, that length is

6^{2} + 3^{2} + 2^{2} = 36 + 9 + 4 = 49 = 7 cm

So a

7

cm rod fits exactly, corner to corner, and a

7.5

cm rod does not fit at all — even though the box's longest edge is only

6

cm. A box holds more length than any of its sides suggests.

That formula is Pythagoras used twice: once across the base to get the face diagonal, then once more upwards. This page covers the first part of the ICSE Class 8 Mathematics chapter on surface area, volume and capacity, and the same habit of taking one dimension at a time runs through all of it.

Formula

What are the surface area and volume formulas for a cuboid and a cube?

For a cuboid of length

l

, breadth

b

and height

h

lateral surface area = 2 h (l + b)

total surface area = 2 (l b + bh + h l)

volume = l \times b \times h

For a cube of edge

a

, all three dimensions are equal, so:

lateral surface area = 4 a^{2}, total surface area = 6 a^{2}, volume = a^{3}

What lateral means. The lateral surface is the four vertical sides only — it leaves out the top and the bottom. That is why it is

2 h (l + b)

: the four walls have heights

h

and their widths add to the perimeter

2 (l + b)

. The total surface adds the two horizontal faces.

Worked example 1. A cuboid measures

12

cm by

4

cm by

3

cm.

volume = 12 \times 4 \times 3 = 144 cm^{3}

TSA = 2 (12 \times 4 + 4 \times 3 + 3 \times 12) = 2 (48 + 12 + 36) = 2 \times 96 = 192 cm^{2}

LSA = 2 \times 3 \times (12 + 4) = 6 \times 16 = 96 cm^{2}

Check the two surface areas against each other. The top and bottom together are

2 \times 48 = 96 cm^{2}

, and

96 + 96 = 192 cm^{2}

. The lateral and total agree, which is a free check on both.

Worked example 2 — a cube. A cube has edge

5

cm.

volume = 125 cm^{3}, TSA = 6 \times 25 = 150 cm^{2}, LSA = 4 \times 25 = 100 cm^{2}

Worked example 3 — working backwards. A cube has volume

343 cm^{3}

. Find its edge and total surface area.

a = 3343 = 7 cm, TSA = 6 \times 49 = 294 cm^{2}

Worked example 4 — from the surface area. A cube has total surface area

486 cm^{2}

6 a^{2} = 486 \Rightarrow a^{2} = 81 \Rightarrow a = 9 cm, volume = 729 cm^{3}

Divide by six before taking the square root. Taking

486

first gives about

22

and is meaningless — the formula has to be undone in reverse order.

Watch the units, which differ between the two quantities. Surface area is in square units and volume in cubic units. A cube of edge

6

cm has TSA

216 cm^{2}

and volume

216 cm^{3}

— the same number with different units, which happens only at edge

6

and is a coincidence rather than a rule.

How do you find the diagonal of a cuboid or a cube?

Apply Pythagoras twice, or use the formula it produces:

diagonal of a cuboid = l^{2} + b^{2} + h^{2}, diagonal of a cube = a 3

Where it comes from. First find the diagonal of the base rectangle:

l^{2} + b^{2}

. That diagonal and the vertical height form a second right-angled triangle whose hypotenuse is the space diagonal:

(l^{2} + b^{2})^{2} + h^{2} = l^{2} + b^{2} + h^{2}

For a cube, all three are

a

, giving

3 a^{2} = a 3

.

Worked example 1. A cuboid measures

6

cm by

3

cm by

2

cm.

diagonal = 36 + 9 + 4 = 49 = 7 cm

Worked example 2. A cuboid measures

12

cm by

4

cm by

3

cm.

diagonal = 144 + 16 + 9 = 169 = 13 cm

Worked example 3 — a room. A room is

8

m long,

4

m wide and

1

m high in a crawl space.

diagonal = 64 + 16 + 1 = 81 = 9 m

Worked example 4 — a cube. A cube of edge

10

cm has

space diagonal = 103 \approx 10 \times 1.732 = 17.32 cm

Compare that with its face diagonal, which uses only two dimensions:

102 \approx 14.14 cm

The face diagonal and the space diagonal are different lengths, and questions ask for both. The face diagonal crosses one square face; the space diagonal goes through the solid. Since

3 > 2

, the space diagonal is always the longer, and it is the longest straight line the solid contains.

Worked example 5 — working backwards. A cuboid has length

6

cm, breadth

4

cm and space diagonal

68

cm. Find its height.

36 + 16 + h^{2} = 68 \Rightarrow h^{2} = 16 \Rightarrow h = 4 cm

The diagonal is never the sum of the edges. For the

12

4

3

cuboid,

12 + 4 + 3 = 19

cm while the diagonal is

13

cm. The diagonal must be longer than the longest edge and shorter than the sum of all three — and checking that your answer sits in that range catches almost every slip.

How do you convert between cubic units and litres?

Cube the length conversion, then remember the one link to capacity.

1 m = 100 cm \Rightarrow 1 m^{3} = 10 0^{3} = 1000000 cm^{3}

1 litre = 1000 cm^{3}, so 1 m^{3} = 1000 litres

Also

1 cm^{3} = 1

millilitre, which makes the litre conversion easy to remember: a litre is a cube of side

10

cm.

**

1 m^{3}

is not

100 cm^{3}

, nor

10000

.** It is a million. Picture a cubic metre filled with centimetre cubes:

100

along,

100

across and

100

high. Volume conversions use the cube of the length factor, just as area conversions used the square — and a wrong power here shifts an answer by a factor of ten thousand.

Worked example 1 — a tank in metres. A tank measures

2

m by

1.5

m by

1

m. Find its capacity in litres.

volume = 2 \times 1.5 \times 1 = 3 m^{3} = 3 \times 1000 = 3000 litres

Worked example 2 — a tank in centimetres. A tank measures

80

cm by

50

cm by

40

cm.

volume = 80 \times 50 \times 40 = 160000 cm^{3} = \frac{160000}{1000} = 160 litres

Worked example 3 — partly filled. A tank

1.2

m by

1

m by

0.8

m is filled with water to a depth of

60

cm. How much water does it hold?

Convert everything to centimetres first:

120

cm by

100

cm, with the water depth of

60

cm as the height:

volume of water = 120 \times 100 \times 60 = 720000 cm^{3} = 720 litres

Use the water's depth, not the tank's height. The tank is

80

cm deep and could hold

960

litres, but the question asked about the water in it. Volume is about the solid; capacity is about what fits inside — and a question about water level is asking for a volume the container is only partly using.

Worked example 4 — cutting a cube up. A cube of edge

12

cm is cut into smaller cubes of edge

3

cm. How many are there?

number = (\frac{12}{3})^{3} = 4^{3} = 64

Check by volume: the big cube is

1728 cm^{3}

and each small one is

27 cm^{3}

, and

\frac{1728}{27} = 64

. Correct.

Dividing the edges gives four, not the answer. The count is

4

along each of three directions, so the number of pieces is

4^{3}

. Answering

4

12

shows the cubing was forgotten, and the volume check settles it immediately.

How do you work out the cost of painting a room or a tank?

Decide exactly which faces are being covered, add their areas, then multiply by the rate.

Worked example 1 — four walls only. A room is

6

m long,

5

m wide and

4

m high. Find the cost of painting the four walls at ₹

25

per square metre.

LSA = 2 h (l + b) = 2 \times 4 \times (6 + 5) = 8 \times 11 = 88 m^{2}

cost = 88 \times 25 = ₹2200

Worked example 2 — walls and ceiling. For the same room, at ₹

40

per square metre.

ceiling = 6 \times 5 = 30 m^{2}, total = 88 + 30 = 118 m^{2}

cost = 118 \times 40 = ₹4720

Read which surfaces the question wants. Four walls is the lateral surface area. Walls and ceiling adds one face. The whole room including floor is the total surface area,

2 (30 + 20 + 24) = 148 m^{2}

. Three different answers from one room, and the only thing that distinguishes them is reading the question carefully.

Worked example 3 — an open tank. A tank

3

m by

2

m by

1.5

m is open at the top. Find the area of sheet metal needed.

An open tank has four walls and a base, so take the total and remove one face:

four walls = 2 \times 1.5 \times (3 + 2) = 3 \times 5 = 15 m^{2}

base = 3 \times 2 = 6 m^{2}, total sheet = 21 m^{2}

Its capacity is

3 \times 2 \times 1.5 = 9 m^{3} = 9000

litres.

Worked example 4 — a door and windows to deduct. The room from example 1 has a door of

2 m \times 1 m

and two windows each

1.5 m \times 1 m

. Find the wall area to be painted.

deduct = 2 \times 1 + 2 \times (1.5 \times 1) = 2 + 3 = 5 m^{2}

painted area = 88 - 5 = 83 m^{2}, cost at ₹25 = ₹2075

Worked example 5 — tiles on a floor. A floor

6

m by

5

m is to be tiled with square tiles of side

50

cm. How many tiles?

Working in metres, each tile is

0.5 \times 0.5 = 0.25 m^{2}

number = \frac{30}{0.25} = 120 tiles

Check by counting along the edges:

\frac{6}{0.5} = 12

tiles along the length and

\frac{5}{0.5} = 10

across, giving

12 \times 10 = 120

. The two methods agree.

Painting uses area, filling uses volume. A question about paint, plaster, sheet metal, tiles or wallpaper is a surface area question; one about water, sand, grain or air is a volume question. Deciding which of the two the question is about, before touching a formula, is the single most valuable habit in this chapter — because the formulas themselves are the easy part.

Exam tip

Exam tip: decide area or volume before choosing a formula

Paint, plaster, sheet and tiles mean surface area. Water, sand and air mean volume. Settle that first.

Read which faces are wanted. For a room

6 \times 5 \times 4

m: four walls

88 m^{2}

, walls plus ceiling

118 m^{2}

, whole room

148 m^{2}

. An open tank is total surface minus one face.

LSA = 2 h (l + b)

TSA = 2 (l b + bh + h l)

V = l bh

. For a cube,

4 a^{2}

6 a^{2}

and

a^{3}

.

Check LSA against TSA: top and bottom together should make up the difference.

Diagonal

= l^{2} + b^{2} + h^{2}

, and

a 3

for a cube. It must be longer than the longest edge and shorter than the sum of the three — a

12 \times 4 \times 3

cuboid gives

13

cm, between

12

and

19

.

Do not confuse the face diagonal

a 2

with the space diagonal

a 3

.

Cube the unit conversion:

1 m^{3} = 1000000 cm^{3}

, and

1

litre

= 1000 cm^{3}

, so

1 m^{3} = 1000

litres.

Convert all lengths to one unit before multiplying, and use the water's depth rather than the tank's height when asked about water.

When a cube is cut up, the count is

(\frac{big}{small})^{3}

— so

12

cm into

3

cm gives

64

, not

4

.

And write square units on areas, cubic units on volumes.

Did you know

Why doubling every edge multiplies the volume by eight

Take a cube of edge

2

cm. Its surface area is

6 \times 4 = 24 cm^{2}

and its volume is

8 cm^{3}

.

Now double every edge, to

4

cm. The surface area becomes

6 \times 16 = 96 cm^{2}

— four times bigger. The volume becomes

64 cm^{3}

— eight times bigger.

The reason is in the powers. Area depends on

a^{2}

, so doubling

a

multiplies it by

2^{2} = 4

. Volume depends on

a^{3}

, so doubling

a

multiplies it by

2^{3} = 8

. Length, area and volume respond to the same change by factors of

2

4

and

8

.

That mismatch has a consequence you can feel. The ratio of surface to volume for a cube of edge

a

\frac{6 a ^{2}}{a ^{3}} = \frac{6}{a}

which shrinks as the cube grows. A large block has relatively little surface for its bulk, and a small one has a great deal.

It is why sugar crystals dissolve faster than a lump of the same total weight, why a cup of tea cools quicker than a full kettle, and why ice is crushed rather than left whole when something needs chilling fast. Breaking a solid into pieces leaves the volume untouched and multiplies the surface — and everything that happens at a surface then happens faster.

The same arithmetic explains the last worked example above. Cutting a

12

cm cube into

3

cm cubes keeps the volume at

1728 cm^{3}

but raises the total surface area from

864 cm^{2}

64 \times 54 = 3456 cm^{2}

— four times as much surface, from the same material.

Key takeaways

Cuboids and cubes: quick revision

- Cuboid:

LSA = 2 h (l + b)

TSA = 2 (l b + bh + h l)

V = l bh

. Cube:

4 a^{2}

6 a^{2}

a^{3}

.
- Lateral means the four vertical sides only; total adds the top and bottom.
- A

12 \times 4 \times 3

cm cuboid has

V = 144 cm^{3}

TSA = 192 cm^{2}

and

LSA = 96 cm^{2}

— and the two faces of

48 cm^{2}

each account for the difference.
- A cube of edge

5

cm has

V = 125 cm^{3}

TSA = 150 cm^{2}

LSA = 100 cm^{2}

.
- Backwards:

V = 343 cm^{3}

gives

a = 7

cm and

TSA = 294 cm^{2}

;

TSA = 486 cm^{2}

gives

a^{2} = 81

, so

a = 9

cm and

V = 729 cm^{3}

. Divide by six before taking the root.
- Space diagonal

= l^{2} + b^{2} + h^{2}

, from Pythagoras used twice. A

6 \times 3 \times 2

cuboid gives

7

cm;

12 \times 4 \times 3

gives

13

cm;

8 \times 4 \times 1

gives

9

m.
- For a cube, the space diagonal is

a 3

and the face diagonal

a 2

— for edge

10

cm,

17.32

cm against

14.14

cm.
- The diagonal is longer than the longest edge and shorter than the sum of all three.
- Cube the conversion:

1 m^{3} = 1000000 cm^{3}

1

litre

= 1000 cm^{3}

1 cm^{3} = 1

mL,

1 m^{3} = 1000

litres.
- A tank

2 \times 1.5 \times 1

m holds

3000

litres; one

80 \times 50 \times 40

cm holds

160

litres.
- Use the water depth: a

120 \times 100

cm tank filled to

60

cm holds

720

litres, not its full

960

.
- A

12

cm cube cut into

3

cm cubes gives

(\frac{12}{3})^{3} = 64

pieces, confirmed by

\frac{1728}{27}

.
- Cost problems: a

6 \times 5 \times 4

m room has four walls of

88 m^{2}

(₹

2200

at ₹

25

), walls plus ceiling

118 m^{2}

(₹

4720

at ₹

40

), whole room

148 m^{2}

.
- Deduct openings: a

2 m^{2}

door and two

1.5 m^{2}

windows leave

83 m^{2}

, costing ₹

2075

.
- An open tank

3 \times 2 \times 1.5

m needs

15 + 6 = 21 m^{2}

of sheet and holds

9000

litres.
- A

6 \times 5

m floor takes

120

tiles of side

50

cm, checked as

12 \times 10

.
- Doubling every edge multiplies area by

4

and volume by

8

, and the surface-to-volume ratio

\frac{6}{a}

shrinks as a solid grows.

Take one room's dimensions and work out all three of its areas — four walls, walls plus ceiling, and the whole room — then decide which a painter would actually charge you for.

What is the longest rod that will fit inside a box?

What are the surface area and volume formulas for a cuboid and a cube?

How do you find the diagonal of a cuboid or a cube?

How do you convert between cubic units and litres?

How do you work out the cost of painting a room or a tank?

Exam tip: decide area or volume before choosing a formula

Why doubling every edge multiplies the volume by eight

Cuboids and cubes: quick revision

Related study notes

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