Surface area and volume of 3D objects (Grade 9) — Mathematics Grades 8–11

Calculating the surface area and volume of prisms and cylinders, and converting SI units.

Gr 9 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 prism or a cylinder.
Calculate the volume of a prism or a cylinder.
Convert between SI units of volume and capacity.
Explain how doubling a dimension changes the volume.

You must be able to

Add the areas of every face, including matching hidden faces, for surface area.
Multiply the area of the base by the height to find volume.
Convert correctly between cubic centimetres, cubic metres and litres.
Explain that doubling the radius of a cylinder makes its volume four times as big.

Traps that cost marks

Forgetting one pair of matching faces of a rectangular prism. A rectangular prism has 3 pairs of matching faces. Count all 6 faces.
Using the diameter instead of the radius to find a cylinder's volume. Always use the radius, half the diameter, when finding the area of the circular base.
Adding the length, width and height instead of multiplying them. Volume of a rectangular prism is length times width times height, multiplied, not added.

Worked example

A closed cylindrical water drum has a radius of 2 m and a height of 3 m. Use pi = 3,14 to calculate its volume.

$V = π r^{2} h$ .
$V = 3, 14 \times 2^{2} \times 3$ .
$V = 3, 14 \times 4 \times 3 = 37, 68$ m³.

Surface area

How much cardboard, paint or wrapping a solid needs, worked out one face at a time.

surface area: The area of all the outside faces of a solid, added together.
face: One flat outside surface of a solid, such as the lid of a box.
prism: A solid with two identical ends joined by rectangles. It stays the same shape all along.
net: The flat pattern you get by unfolding a solid. It shows every face at once.
curved surface: The rounded outside of a cylinder. Unrolled, it opens out into a rectangle.

Surface area is the area of the outside of a solid, so it is measured in square units. If a question talks about paint, wrapping paper or cardboard, it is asking for surface area.

Unfold the solid into a net and count every face. A rectangular prism has $6$ faces in $3$ matching pairs. Work out three areas, then double the total.

A triangular prism has $5$ faces: TWO triangles and three rectangles. The triangles use the height measured at a right angle to the base. The rectangles use the length of the prism, which is a different measurement.

A closed cylinder unrolls into two circles and one rectangle. That rectangle is as long as the distance round the circle, which is $2 π r$ , and as wide as the height. So the curved surface is $2 π r h$ .

Now add the two circular ends. Each is $π r^{2}$ , so the two come to $2 π r^{2}$ . A tin with no lid has only one end, so read the question before you add.

Rules to remember

rectangular prism: $S A = 2 (l b + l h + bh)$
closed cylinder: $S A = 2 π r^{2} + 2 π r h$

Examples

A closed cardboard box measures

10

cm by

6

cm by

4

cm. Find its surface area.

Worked answer

Three different faces: $10 \times 6 = 60$ , $10 \times 4 = 40$ and $6 \times 4 = 24$ .
Add those three: $60 + 40 + 24 = 124$ .
Every face has a matching one opposite it, so double the total.
$124 \times 2 = 248$ square centimetres.

Answer: $248$ cm $^{2}$

Stopping at $124$ would pay for only three of the six faces.

A wooden doorstop is a triangular prism

10

cm long. Its end is a right-angled triangle with sides

3

cm,

4

cm and

5

cm. Find its surface area.

Worked answer

One triangular end: $\frac{1}{2} \times 3 \times 4 = 6$ square centimetres.
There are two ends, so that is $12$ .
The three rectangles are all $10$ cm long: $3 \times 10 = 30$ , $4 \times 10 = 40$ and $5 \times 10 = 50$ .
Rectangles: $30 + 40 + 50 = 120$ , so the total is $12 + 120 = 132$ .

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

There are TWO triangles, and the rectangles are as long as the prism itself.

A closed tin has a diameter of

10

cm and a height of

7

cm. Take

π

3, 14

and find its surface area.

Worked answer

The diameter is $10$ cm, so the radius is $5$ cm.
The two ends: $2 \times 3, 14 \times 5^{2} = 157$ .
Curved surface, first part: $2 \times 3, 14 \times 5 = 31, 4$ .
Then $31, 4 \times 7 = 219, 8$ .
$157 + 219, 8 = 376, 8$ square centimetres.

Answer: $376, 8$ cm $^{2}$

A closed tin has a base AND a lid, so both circles are counted.

Traps

Forgetting one pair of matching faces of a rectangular prism. A rectangular prism has 3 pairs of matching faces. Count all 6 faces.
Working out the volume when the question asked for surface area. Surface area adds up flat areas, so the answer comes out in square units.
Using the triangle's own height as the length of a rectangular face. The rectangles are as long as the prism. That height belongs to the triangle only.
Leaving out one or both circular ends of a closed cylinder. A closed tin has a base and a lid, so add $2 π r^{2}$ for the pair.

Learn and practise “Surface area” in the app →

Volume and capacity

How much space is inside a solid, how many litres that comes to, and why doubling is never just doubling.

volume: The amount of space inside a solid, counted in cubes.
cubic centimetre: A cube that is $1$ cm along every edge. Volume is counted in these.
capacity: How much liquid a container holds, given in millilitres, litres or kilolitres.
cross-section: The flat shape you would see if you sliced straight through a prism.

Volume is the space inside a solid, so its units are cubed. A cubic centimetre is a cube $1$ cm along each edge. Surface area is flat, with squared units, so check your unit.

For any prism, volume is the cross-section area times the length. For a rectangular prism that is length times breadth times height. Those three are MULTIPLIED, never added.

A triangular prism works the same way. Find the triangle's area first, keeping the half, then multiply by the length of the prism. That length is not the triangle's height.

A cylinder is a prism whose cross-section is a circle. The base area is $π r^{2}$ , so the volume is $π r^{2} h$ . Halve the diameter first, and use the base AREA, not the distance round it.

Volume units use the length factor three times over. $1$ cm is $10$ mm, so $1$ cm $^{3}$ is $1000$ mm $^{3}$ . Capacity is a separate list: $1000$ cm $^{3}$ is $1$ litre, and $1$ m $^{3}$ is a kilolitre.

Rules to remember

rectangular prism: $V = l \times b \times h$
cylinder: $V = π r^{2} h$
$1000$ cm $^{3}$ is $1$ litre, and $1$ m $^{3}$ is $1$ kilolitre

Examples

A rectangular water tank measures

2

m by

1

m by

1

m. Find its volume, then its capacity in litres.

Worked answer

Multiply the three measurements: $2 \times 1 \times 1 = 2$ .
So the volume is $2$ cubic metres.
One cubic metre holds $1$ kilolitre, which is $1000$ litres.
$2 \times 1000 = 2000$ , so the tank holds $2000$ litres.

Answer: $2$ m $^{3}$ , which is $2000$ litres

Adding the three would give $4$ , which is not a volume at all.

A chocolate box is a triangular prism

20

cm long. Its triangular end has a base of

6

cm and a height of

4

cm. Find its volume.

Worked answer

The triangle first: $\frac{1}{2} \times 6 \times 4 = 12$ square centimetres.
That triangle is the cross-section of the prism.
Multiply by the length of the box: $12 \times 20 = 240$ .
The volume is $240$ cubic centimetres.

Answer: $240$ cm $^{3}$

The $20$ cm is the length of the box, not the height of its triangle.

A paint tin has a radius of

5

cm and a height of

10

cm. Take

π

3, 14

. Find its volume. What would doubling the radius do?

Worked answer

Base area: $3, 14 \times 5^{2} = 78, 5$ square centimetres.
Volume: $78, 5 \times 10 = 785$ cubic centimetres.
The radius is squared, so doubling it multiplies by $2 \times 2 = 4$ .
The new volume would be $785 \times 4 = 3140$ cubic centimetres.

Answer: $785$ cm $^{3}$ , and four times that if the radius doubles

Doubling a radius does not double a volume, because the radius is squared.

Traps

Adding the length, width and height instead of multiplying them. Volume of a rectangular prism is length times width times height, multiplied, not added.
Leaving out the half when working out the triangle end of a prism. The cross-section is a triangle, so its area is half of the base times the height.
Using the diameter instead of the radius to find a cylinder's volume. Always use the radius, half the diameter, when finding the area of the circular base.
Saying that one cubic metre holds one litre. One cubic metre holds $1$ kilolitre, and that is $1000$ litres.

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

Grade 9 Surface area and volume of 3D objects

What gets asked

You must be able to

Traps that cost marks

Worked example

Surface area

Rules to remember

Examples

Traps

Volume and capacity

Rules to remember

Examples

Traps

About this material