About Volume and Surface Area
Volume is the space a solid occupies and surface area is the amount of material needed to cover it. They are built from the same dimensions but use different formulas, and the most common mistake in the topic is to work out one when the question asked for the other.
What you need to understand
- Volume is measured in cubic units and surface area in square units, so reading the units of the answer catches most errors.
- The volume of a cuboid is length x breadth x height, while its total surface area is the sum of the areas of its six faces.
- A cube has six identical faces, so its total surface area is six times the square of the edge.
- For a cylinder, pi times the radius squared times the height gives the volume, while the curved surface is the circumference times the height.
- Doubling every dimension of a solid multiplies the volume by eight but the surface area by only four.
- A cylinder with the same radius and height as a cone holds exactly three times as much.
Formulas to remember
How to work through these questions
- Decide first whether the question wants a volume in cubic units or an area in square units, then write down the matching formula.
- For a cylinder, check that you have the radius rather than the diameter, and square the radius before multiplying by the height.
- Take pi as 22/7 whenever the radius is a multiple of seven, which is true of almost every exam question on cylinders and spheres.
- For a cuboid, remember that the volume uses a product of three lengths while the surface area uses pairs of faces, and keep the two formulas apart.
- Check the size of the answer, because a volume is always larger than the matching surface area for the same solid.
Mistakes that cost marks
- Giving the volume when the question asked for the surface area, or the reverse.
- Using the diameter as though it were the radius.
- Counting only four faces of a cube instead of six.
- Adding the three edges together instead of multiplying them when finding a volume.
- Dropping the factor of one third in the volume of a cone or a pyramid.
Worked example
Find the volume of a cylinder of radius 7 cm and height 10 cm, taking pi as 22/7.
- The volume of a cylinder is pi x r^2 x h.
- Pi x r^2 = (22/7) x 7 x 7 = 22 x 7 = 154.
- Multiply by the height: 154 x 10 = 1540 cubic cm.
- Check: a base area of 154 square cm multiplied by a height of 10 cm gives 1540, and the units are cubic, as a volume must be.
Answer: 1540 cubic cm
Practice questions with answers
A few Volume and Surface Area questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
A box measures 12 cm by 7 cm by 8 cm. What is its volume?
Answer: Option A — with explanation
Volume of a cuboid = length x breadth x height.
= 12 x 7 x 8 = 672 cubic cm.
Common mistakes
- added the areas of the three faces instead of multiplying all three sides: 236
- gave the total surface area instead of the volume: 472
- added the three dimensions: 27
Question 2
Find the volume of a cylinder of radius 28 cm and height 9 cm. (Take $\pi = \dfrac{22}{7}$)
-
A
22176
-
B
1584
-
C
792
-
D
11088
Answer: Option A — with explanation
Volume of a cylinder = $\pi r^{2} h$.
= $\dfrac{22}{7} \times 28^{2} \times 9$ = 22176 cubic cm.
Common mistakes
- halved the volume as though the solid were a cone: 11088
- used the radius once instead of squaring it: 792
- gave the curved surface area instead of the volume: 1584
Question 3
Calculate the volume of a cuboid of dimensions 13 cm x 9 cm x 8 cm.
Answer: Option B — with explanation
Volume of a cuboid = length x breadth x height.
= 13 x 9 x 8 = 936 cubic cm.
Common mistakes
- added the areas of the three faces instead of multiplying all three sides: 293
- gave the total surface area instead of the volume: 586
- added the three dimensions: 30
Question 4
A cylinder has a radius of 21 cm and a height of 18 cm. What is its volume? (Take $\pi = \dfrac{22}{7}$)
-
A
2376
-
B
1188
-
C
24948
-
D
12474
Answer: Option C — with explanation
Volume of a cylinder = $\pi r^{2} h$.
= $\dfrac{22}{7} \times 21^{2} \times 18$ = 24948 cubic cm.
Common mistakes
- used the radius once instead of squaring it: 1188
- halved the volume as though the solid were a cone: 12474
- gave the curved surface area instead of the volume: 2376
Question 5
Calculate the volume of a cuboid of dimensions 11 cm x 7 cm x 9 cm.
Answer: Option A — with explanation
Volume of a cuboid = length x breadth x height.
= 11 x 7 x 9 = 693 cubic cm.
Common mistakes
- added the areas of the three faces instead of multiplying all three sides: 239
- gave the total surface area instead of the volume: 478
- added the three dimensions: 27
Frequently asked questions
What is the difference between volume and surface area?
Volume measures the space inside the solid and is given in cubic units. Surface area measures the skin around the outside and is given in square units. A solid has both, and one does not determine the other.
Why does a cube use six times the square of the edge?
Because a cube has six identical square faces and each face has an area of the edge multiplied by itself.
How do I know when to use 22/7 for pi?
Use 22/7 when the radius or the diameter is a multiple of seven, because the seven cancels and the answer comes out as a whole number. Otherwise use 3.14.
What happens to the volume if every dimension is doubled?
The volume becomes eight times as large, because each of the three dimensions is doubled and two cubed is eight. The surface area becomes only four times as large.
