Surface area and volume of 3D objects (Grade 8) — Mathematics Grades 8–11

Calculating surface area, volume and capacity of prisms, and converting units.

Gr 8 Term 3 — Surface area and volume of 3D objectsTerm 3

Practise Surface area and volume of 3D objects in the app →

What gets asked

Calculate the surface area of a cube, rectangular prism or triangular prism.
Calculate the volume or capacity of a prism.
Convert between units of volume and capacity.
Solve a problem in context involving surface area, volume or capacity.

You must be able to

Add up the area of every face, including hidden ones, for surface area.
Multiply length, width and height for the volume of a prism.
Keep surface area and volume separate: square units, then cubic units.
Convert between ml, litres and kilolitres, and between mm³, cm³ and m³.

Traps that cost marks

Multiplying the three dimensions of a prism when surface area is what is asked for. Surface area adds up the area of every face. Volume multiplies length, width and height.
Adding the length, width and height of a prism instead of multiplying them. Volume is length $\times$ width $\times$ height, not their sum.
Using the linear conversion factor for volume, e.g. treating $1 m^{3}$ as $100 cm^{3}$ . Volume units are cubed, so the conversion factor is cubed too: $1 m^{3} = 1000000 cm^{3}$ .
Leaving a capacity answer in cm³ when the question needs litres. Check the units the question asks for. $1000 cm^{3} = 1$ litre.

Worked example

A water tank shaped like a rectangular prism is

2

m long,

1

m wide and

1

m high. Find its volume.

Volume of a rectangular prism $=$ length $\times$ width $\times$ height.
$2 \times 1 \times 1$ .
Volume $= 2$ m³.

Surface area of prisms

A box is flat before it is a box. Unfold it and every side is there to measure.

face: One of the flat surfaces of a solid. A cube has $6$ of them.
prism: A solid with two identical ends joined by rectangles. A shoebox and a tent are both prisms.
net: The flat shape you get by unfolding a solid so that every face lies open.
surface area: The area of every face of a solid, added together.

Surface area is what it takes to cover a solid: the wrapping paper, the tin, the paint. So work out the area of each face and add them all. The answer is an area, so its unit is squared.

A rectangular prism has $6$ faces in three matching pairs: front and back, top and bottom, and the two sides. Work out one of each pair, add those three, then double. A cube is easier still. Its $6$ faces are equal squares, so find one and multiply by $6$ .

A triangular prism has $5$ faces. Two are identical triangles, one at each end. The other three are rectangles joining them. Every one of those rectangles is as long as the prism, and their widths are the three sides of the triangle.

A drawing of a solid hides the faces at the back. Draw its net instead, or count the faces in pairs, and none of them goes missing.

Rules to remember

cube: surface area $= 6 s^{2}$ , where $s$ is one edge
rectangular prism: surface area $= 2 (l b + l h + bh)$

Examples

A wooden block is a cube with edges of

5

cm. Find its surface area.

Worked answer

One face is a square: $5 \times 5 = 25$ .
A cube has $6$ equal faces.
$6 \times 25 = 150$ .
The surface area is $150$ cm $^{2}$ .

Answer: $150$ cm $^{2}$

Every face of a cube is the same, so one area does for all six.

A cereal box is

30

cm tall,

20

cm wide and

6

cm deep. How much cardboard covers it?

Worked answer

Front: $30 \times 20 = 600$ .
Side: $30 \times 6 = 180$ .
Top: $20 \times 6 = 120$ .
Each of those has a partner behind it, so add them and double.
$600 + 180 + 120 = 900$ , and $2 \times 900 = 1800$ .

Answer: $1800$ cm $^{2}$

Only three faces were worked out, because a box has three matching pairs.

A wooden doorstop is a triangular prism

10

cm long. Its end triangle has sides of

3

cm,

4

cm and

5

cm, with the right angle between the

3

cm and

4

cm sides.

Worked answer

One triangle: $\frac{1}{2} \times 3 \times 4 = 6$ .
There is an end at each side, so $2 \times 6 = 12$ .
The three rectangles are each $10$ cm long.
Their widths add up: $3 + 4 + 5 = 12$ , so together they cover $12 \times 10 = 120$ .
Total: $12 + 120 = 132$ .

Answer: $132$ cm $^{2}$

The $5$ cm slant is the width of a rectangle, not the height of the triangle.

Traps

Adding only the three faces you can see in the drawing. A closed box has six faces. Each one you can see has a partner the same size behind it.
Multiplying the three dimensions of a prism when surface area is what is asked for. Surface area adds up the area of every face. Volume multiplies length, width and height.
Counting one triangular end instead of two. A triangular prism has an end at each side. Work out one triangle and double it.
Using the slanted side of the triangle as the length of a rectangular face. Every rectangle is as long as the whole prism. The slanted side is only its width.

Learn and practise “Surface area of prisms” in the app →

Volume, capacity and problems

How many litres a tank holds starts out as three measurements multiplied together.

volume: The amount of space a solid takes up. It is measured in cubes.
capacity: The amount a container can hold, measured in millilitres, litres or kilolitres.
cross-section: The shape you see if you cut straight across a prism. It is the same all the way along.

Volume is how much space a solid fills. For a rectangular prism, multiply length by width by height. Multiply them; never add them. A cube is three equal edges multiplied together. The unit is cubed: mm $^{3}$ , cm $^{3}$ or m $^{3}$ .

Every prism works the same way. Find the area of its cross-section, then multiply by the length. For a triangular prism the cross-section is the triangle at the end, so its height is measured across that end, not along the prism.

Volume units are cubed, so cube the factor: $1 cm^{3} = 1000 mm^{3}$ and $1 m^{3} = 1000000 cm^{3}$ . Capacity has units of its own. A space of $1$ cm $^{3}$ holds $1$ millilitre, so $1000$ cm $^{3}$ is $1$ litre, and $1000$ litres is $1$ kilolitre.

Surface area and volume do not rise together. Two containers can hold exactly the same and still need different amounts of card. Double every edge of a box and its surface area becomes $4$ times as big, while its volume becomes $8$ times as big.

In a word problem, ask what is wanted. Covering, wrapping or painting means surface area. Filling means volume or capacity. Then check the unit the answer must be in, and convert at the end.

Rules to remember

rectangular prism: volume $= l \times b \times h$
any prism: volume $=$ cross-section area $\times$ length

Examples

A rectangular water tank is

80

cm long,

50

cm wide and

40

cm deep. How many litres does it hold?

Worked answer

Volume: $80 \times 50 \times 40$ .
$80 \times 50 = 4000$ , and $4000 \times 40 = 160000$ .
So the volume is $160000$ cm $^{3}$ .
A space of $1000$ cm $^{3}$ holds $1$ litre, so divide by $1000$ .
$160000 \div 1000 = 160$ .

Answer: $160$ litres

The volume came out in cm $^{3}$ , and the question asked for litres.

A cattle trough is a triangular prism

200

cm long. Its end is a triangle

60

cm across the top and

40

cm deep, with two equal sloping sides. How many litres does it hold?

Worked answer

The cross-section is the triangle at the end.
Its area: $\frac{1}{2} \times 60 \times 40 = 1200$ .
Volume: $1200 \times 200 = 240000$ .
$240000 \div 1000 = 240$ .

Answer: $240$ litres

The $40$ cm is measured across the end. The $200$ cm runs along the prism.

Two boxes are made. One is a cube with edges of

2

cm. The other is

1

cm by

1

cm by

8

cm. Do they hold the same, and do they need the same card?

Worked answer

Cube volume: $2 \times 2 \times 2 = 8$ .
Long box volume: $1 \times 1 \times 8 = 8$ , which is the same.
One face of the cube is $2 \times 2 = 4$ , so its card is $6 \times 4 = 24$ .
The long box needs $2 \times (1 + 8 + 8) = 34$ .
Same volume, more card.

Answer: both $8$ cm $^{3}$ ; card $24$ cm $^{2}$ against $34$ cm $^{2}$

Equal volumes do not force equal surface areas. A long thin box has more outside.

Traps

Adding the length, width and height of a prism instead of multiplying them. Volume is length $\times$ width $\times$ height, not their sum.
Using the length of the prism as the height of its end triangle. The height of that triangle is measured across the end. The length runs along the prism.
Using the linear conversion factor for volume, e.g. treating $1 m^{3}$ as $100 cm^{3}$ . Volume units are cubed, so the conversion factor is cubed too: $1 m^{3} = 1000000 cm^{3}$ .
Leaving a capacity answer in cm $^{3}$ when the question needs litres. Check the units the question asks for. $1000 cm^{3} = 1$ litre.

Learn and practise “Volume, capacity and problems” in the app →

Grade 8 Surface area and volume of 3D objects

What gets asked

You must be able to

Traps that cost marks

Worked example

Surface area of prisms

Rules to remember

Examples

Traps

Volume, capacity and problems

Rules to remember

Examples

Traps

About this material