Volume and surface area of solids

Q: How do I convert volume to litres?

1 cm³ = 1 millilitre, and 1000 cm³ = 1 litre. Also 1 m³ = 1000 litres.

Volume is the space a solid takes up, measured in cubic units. Surface area is the total area of all its faces, measured in square units. Each solid has its own pair of formulas.

Formulas

Cuboid (l × b × h)

V = l \times b \times h TS A = 2 (l b + bh + h l)

Cube (side a)

V = a^{3} TS A = 6 a^{2}

Cylinder (radius r, height h)

V = π r^{2} h CS A = 2 π r h TS A = 2 π r (r + h)

Cone (radius r, height h)

V = \frac{1}{3} π r^{2} h

Worked examples

Example 1. Find the volume and total surface area of a cuboid 8 cm long, 5 cm wide and 3 cm high.

$V = 8 \times 5 \times 3 = 120$ .
$TS A = 2 (8 \times 5 + 5 \times 3 + 3 \times 8) = 2 (40 + 15 + 24) = 158$ .

Answer:

V = 120 cm^{3}

, TSA =

158 cm^{2}

Example 2. Find the volume and total surface area of a cube of side 6 cm.

$V = 6^{3} = 216$ .
$TS A = 6 \times 6^{2} = 216$ .

Answer:

V = 216 cm^{3}

, TSA =

216 cm^{2}

Example 3. A cylinder has radius 7 cm and height 10 cm. Find its volume and curved surface area (take

π = \frac{22}{7}

$V = \frac{22}{7} \times 7^{2} \times 10 = 22 \times 7 \times 10 = 1540$ .
$CS A = 2 \times \frac{22}{7} \times 7 \times 10 = 440$ .

Answer:

V = 1540 cm^{3}

, CSA =

440 cm^{2}

Example 4. A tank is 2 m long, 1.5 m wide and 1 m deep. How many litres of water can it hold?

$V = 2 \times 1.5 \times 1 = 3 m^{3}$ .
$1 m^{3} = 1000$ litres, so $3 \times 1000 = 3000$ .

Answer: 3000 litres

Common mistakes

Writing square units for volume, or cubic units for area.
Using the diameter in place of the radius in the cylinder and cone formulas.
Forgetting the $\frac{1}{3}$ in the volume of a cone.

Practice questions with answers

1. What is the formula for the volume of a cuboid with length l, breadth b and height h?

A $l x b x h$
B2(lb + bh + hl)
Cl + b + h
D $\frac{1}{2} x l x b x h$

Show answer

Answer: A)

l x b x h

Volume of a cuboid = length

x

breadth

x

height =

l x

x

2. The total surface area of a cube of edge a is given by which formula?

A $6 a^{2}$
B $4 a^{2}$
C $a^{3}$
D $6 a$

Show answer

Answer: A)

6 a^{2}

A cube has 6 equal square faces, each of area

a^{2}

. Total surface area =

6 a^{2}

3. Which formula gives the curved surface area of a cylinder with base radius r and height h?

Aπ $r^{2}$ h
B2πrh
Cπrh
D2πr(r + h)

Show answer

Answer: B) 2πrh

Unrolled, the curved surface is a rectangle of length 2π

r

(the base circumference) and breadth

h

, so its area is 2π

r h

. π

r^{2}

h

is the volume, and 2π

r (r + h)

is the total surface area.

4. The volume of a cone is given by which formula? (r = base radius, h = height)

A $\frac{1}{3} p i r^{2} h$
B $p i r^{2} h$
C $\frac{2}{3} p i r^{2} h$
D $\frac{1}{3} p i r h^{2}$

Show answer

Answer: A)

\frac{1}{3} p i r^{2} h

A cone is one-third of a cylinder of the same base and height, so Volume =

\frac{1}{3}

r^{2}

5. A cuboidal tank 2 m

x

1.5 m

x

1 m is being filled by a pipe delivering water at 50 litres per minute.
How long (in minutes) will it take to fill the tank completely?

A60
B54
C120
D66

Show answer

Answer: A) 60

Volume =

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

L. Time = 3000 / 50 = 60 minutes.

6. A cubical tank of edge 1.2 m is open at the top.
It is to be lined with tiles on the inside (4 walls and the base) at Rs 250 per sq m.
Find the total cost of tiling.

ARs 1800
BRs 2160
CRs 1440
DRs 2592

Show answer

Answer: A) Rs 1800

Open tank = 5 faces =

5 a^{2} = 5 \times 1.44 = 7.2

sq m. Cost =

7.2 \times 250

= Rs 1800.

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Frequently asked questions

What is the difference between volume and surface area?

Volume is how much space is inside the solid (cubic units). Surface area is the total area of the outside (square units).

What is the difference between CSA and TSA?

Curved surface area is the curved part only (the side of a cylinder). Total surface area adds the flat faces, such as the two circles of a cylinder.

How do I convert volume to litres?

$1 cm^{3} = 1$ millilitre, and $1000 cm^{3} = 1$ litre. Also $1 m^{3} = 1000$ litres.

Why is a cone one third of a cylinder?

A cone with the same base and height as a cylinder holds exactly one third as much, so $V_{cone} = \frac{1}{3} π r^{2} h$ .

Volume and surface area of solids

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

Related topics