Sec 1 · G3 (Express)

Volume and surface area

Practise volume and surface area with step-by-step help. Free account.

1. Cubes, cuboids and prisms

Solid	Volume	Total surface area
cube, side $l$	$l^{3}$	$6l^{2}$
cuboid	$l \times b \times h$	$2(lb + lh + bh)$
prism	area of cross-section $\times$ length	$2 \times$ area of cross-section $+$ perimeter of cross-section $\times$ length
cylinder	$\pi r^{2}h$	$2\pi r^{2} + 2\pi rh$

Cuboid 8 cm by 3 cm by 5 cm

\displaystyle V = 8 \times 3 \times 5

e.g.

\displaystyle 8 \times 3 \times 5

\displaystyle = 120

$120\text{ cm}^{3}$
surface area: $2(8 \times 3 + 8 \times 5 + 3 \times 5)$ , which is $158\text{ cm}^{2}$

Triangular prism: cross-section a right triangle 3 cm by 4 cm, length 7 cm

\displaystyle V = \dfrac{1}{2} \times 3 \times 4 \times 7

e.g.

\displaystyle \dfrac{1}{2} \times 3 \times 4 \times 7

\displaystyle = 42

$42\text{ cm}^{3}$

📖 Worked example

The diagram shows a tent in the shape of a triangular prism. Find its volume.

Step 1 of 2

Area of the triangle (the cross-section)

\text{Area} = \dfrac{1}{2} \times 6 \times 4

= 12

cm²

📖 Worked example

The diagram shows a wooden chest in the shape of a cuboid. Find its total surface area.

Step 1 of 1

Six faces in three equal pairs: top and bottom, front and back, the two ends

\text{TSA} = 2(9 \times 4) + 2(9 \times 7) + 2(4 \times 7)

= 254

cm²

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2. Cylinders

Volume, radius 5 cm and height 12 cm

\displaystyle V = \pi r^{2}h

e.g.

\displaystyle \pi(5)^{2}(12)

\displaystyle = 300\pi

\displaystyle = 942\text{ (3 s.f.)}

Curved surface only: no circles

\displaystyle 2\pi rh

e.g.

\displaystyle 2\pi(5)(12)

\displaystyle = 120\pi

\displaystyle = 377\text{ (3 s.f.)}

📖 Worked example

The diagram shows a cylindrical candle. Find its volume, correct to 3 significant figures.

Step 1 of 1

Volume of a cylinder = πr²h: round at the end

V = \pi r^2 h

= \pi(3)^2(4)

= 113.0973\ldots

= 113

cm³ (3 s.f.)

📖 Worked example

A closed cylinder has a radius of $3$ cm and a total surface area of $60\pi$ cm². Find its height, $h$ cm.

Step 1 of 2

Total surface area = two circular ends + the curved surface

2\pi r^2 + 2\pi rh = 60\pi

2\pi(3)^2 + 2\pi(3)h = 60\pi

📖 Worked example

$6$ identical tins of paint, each of radius $2$ cm and height $8$ cm, stand upright in a closed rectangular box: $3$ along its length and $2$ across its width. They fit exactly, touching each other, the sides, the top and the bottom of the box. Find the volume of air space in the box, correct to 3 significant figures.

Step 1 of 3

The box: each can takes up a diameter (2r) of the length and of the width

Length

= 3 \times 4

= 12

cm, width

= 2 \times 4

= 8

\text{Box} = 12 \times 8 \times 8

\phantom{\text{Box}} = 768

cm³

📖 Worked example

The diagram shows a water trough in the shape of a prism of length $80$ cm. It is empty, and water flows in at a constant rate of $3$ litres per minute. It is full after $12$ minutes. Find the area of its cross-section, in cm².

Step 1 of 3

Volume of water = rate × time

\text{Volume} = 3 \times 12

\phantom{\text{Volume}} = 36

litres

📖 Worked example

A rectangular sheet of paper, $22$ cm by $15$ cm, is rolled up so that its two edges of length $15$ cm just meet, without overlapping. It forms the curved surface of a cylinder of height $15$ cm. Find the radius of the cylinder, correct to 3 significant figures.

Step 1 of 1

The 22 cm edge goes round the cylinder: it is the circumference

2\pi r = 22

r = \dfrac{22}{2\pi}

= 3.5014\ldots

= 3.50

cm (3 s.f.)

📖 Worked example

The diagram shows a wooden box in the shape of a cuboid, with a lid. Its outside, including the lid, is painted with paint that costs $4 per m². Find the cost of the paint needed.

Step 1 of 3

Six faces in three equal pairs

\text{Area} = 2(50 \times 40) + 2(50 \times 30) + 2(40 \times 30)

= 9400

cm²

📖 Worked example

$36$ identical 1 cm cubes are put together to make a solid cuboid. The top face of the cuboid has a perimeter of $10$ cm, and every edge of the cuboid is longer than $1$ cm. Find the height of the cuboid.

Step 1 of 3

The top face: 2 × (length + width) = perimeter

2(\text{length} + \text{width}) = 10

, so

\text{length} + \text{width} = 5

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3. Units of volume

Volume	Equals
$1\text{ m}^{3}$	$1\,000\,000\text{ cm}^{3}$
$1\text{ litre}$	$1000\text{ cm}^{3}$
$1\text{ m}^{3}$	$1000\text{ litres}$
$1\text{ ml}$	$1\text{ cm}^{3}$

📖 Worked example

A cuboid container measures $120$ cm by $50$ cm by $40$ cm. Find its capacity, in litres.

Step 1 of 2

Volume of the tank in cm³

V = 120 \times 50 \times 40

= 240\,000

cm³

Practise this →

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Sec 1 · G3 (Express)

Volume and surface area

Practise volume and surface area with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Cubes, cuboids and prisms

Solid	Volume	Total surface area
cube, side $l$	$l^{3}$	$6l^{2}$
cuboid	$l \times b \times h$	$2(lb + lh + bh)$
prism	area of cross-section $\times$ length	$2 \times$ area of cross-section $+$ perimeter of cross-section $\times$ length
cylinder	$\pi r^{2}h$	$2\pi r^{2} + 2\pi rh$

Cuboid 8 cm by 3 cm by 5 cm

\displaystyle V = 8 \times 3 \times 5

e.g.

\displaystyle 8 \times 3 \times 5

\displaystyle = 120

$120\text{ cm}^{3}$
surface area: $2(8 \times 3 + 8 \times 5 + 3 \times 5)$ , which is $158\text{ cm}^{2}$

Triangular prism: cross-section a right triangle 3 cm by 4 cm, length 7 cm

\displaystyle V = \dfrac{1}{2} \times 3 \times 4 \times 7

e.g.

\displaystyle \dfrac{1}{2} \times 3 \times 4 \times 7

\displaystyle = 42

$42\text{ cm}^{3}$

📖 Worked example

The diagram shows a tent in the shape of a triangular prism. Find its volume.

Step 1 of 2

Area of the triangle (the cross-section)

\text{Area} = \dfrac{1}{2} \times 6 \times 4

= 12

cm²

📖 Worked example

The diagram shows a wooden chest in the shape of a cuboid. Find its total surface area.

Step 1 of 1

Six faces in three equal pairs: top and bottom, front and back, the two ends

\text{TSA} = 2(9 \times 4) + 2(9 \times 7) + 2(4 \times 7)

= 254

cm²

Practise this →

2. Cylinders

Volume, radius 5 cm and height 12 cm

\displaystyle V = \pi r^{2}h

e.g.

\displaystyle \pi(5)^{2}(12)

\displaystyle = 300\pi

\displaystyle = 942\text{ (3 s.f.)}

Curved surface only: no circles

\displaystyle 2\pi rh

e.g.

\displaystyle 2\pi(5)(12)

\displaystyle = 120\pi

\displaystyle = 377\text{ (3 s.f.)}

📖 Worked example

The diagram shows a cylindrical candle. Find its volume, correct to 3 significant figures.

Step 1 of 1

Volume of a cylinder = πr²h: round at the end

V = \pi r^2 h

= \pi(3)^2(4)

= 113.0973\ldots

= 113

cm³ (3 s.f.)

📖 Worked example

A closed cylinder has a radius of $3$ cm and a total surface area of $60\pi$ cm². Find its height, $h$ cm.

Step 1 of 2

Total surface area = two circular ends + the curved surface

2\pi r^2 + 2\pi rh = 60\pi

2\pi(3)^2 + 2\pi(3)h = 60\pi

📖 Worked example

Step 1 of 3

The box: each can takes up a diameter (2r) of the length and of the width

Length

= 3 \times 4

= 12

cm, width

= 2 \times 4

= 8

\text{Box} = 12 \times 8 \times 8

\phantom{\text{Box}} = 768

cm³

📖 Worked example

Step 1 of 3

Volume of water = rate × time

\text{Volume} = 3 \times 12

\phantom{\text{Volume}} = 36

litres

📖 Worked example

Step 1 of 1

The 22 cm edge goes round the cylinder: it is the circumference

2\pi r = 22

r = \dfrac{22}{2\pi}

= 3.5014\ldots

= 3.50

cm (3 s.f.)

📖 Worked example

The diagram shows a wooden box in the shape of a cuboid, with a lid. Its outside, including the lid, is painted with paint that costs $4 per m². Find the cost of the paint needed.

Step 1 of 3

Six faces in three equal pairs

\text{Area} = 2(50 \times 40) + 2(50 \times 30) + 2(40 \times 30)

= 9400

cm²

📖 Worked example

Step 1 of 3

The top face: 2 × (length + width) = perimeter

2(\text{length} + \text{width}) = 10

, so

\text{length} + \text{width} = 5

Practise this →

3. Units of volume

Volume	Equals
$1\text{ m}^{3}$	$1\,000\,000\text{ cm}^{3}$
$1\text{ litre}$	$1000\text{ cm}^{3}$
$1\text{ m}^{3}$	$1000\text{ litres}$
$1\text{ ml}$	$1\text{ cm}^{3}$

📖 Worked example

A cuboid container measures $120$ cm by $50$ cm by $40$ cm. Find its capacity, in litres.

Step 1 of 2

Volume of the tank in cm³

V = 120 \times 50 \times 40

= 240\,000

cm³

Practise this →

Try a question: Volume and surface area

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