Volume of a cuboid, prism and cylinder
Volume is how much space a solid takes up, in cm³ or litres. The volume of a cuboid is length · width · height. Prism and cylinder: the area of the base · the height. 1 dm³ = 1 litre = 1 000 cm³.
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Cuboid and cube
Volume counts how many cubes of 1 cm · 1 cm · 1 cm fit inside. One such cube is 1 cm³.
A cuboid of 5 · 4 · 3 cm has 20 cubes in the bottom layer and three layers.
V = l · b · h = 5 · 4 · 3 = 60 cm³
A cube with side 4 cm has a volume of 4 · 4 · 4 = 4³ = 64 cm³.
Base area times height
A solid that is equally thick all the way up is layer upon layer of the base area B.
V = B · h
| Solid | Base area B | Volume |
|---|---|---|
| Cuboid | l · b | l · b · h |
| Cube | s · s | s · s · s = s³ |
| Prism | The area of the base shape, for example a triangle: b · h ÷ 2 | B · height |
| Cylinder | A circle: π · r² | π · r² · h |
| Cone and pyramid (Year 9) | Cone: a circle. Pyramid: a polygon. | B · h ÷ 3 |
| Sphere (Year 9) | None, the sphere is round in every direction | 4 · π · r³ ÷ 3 |
A cone or pyramid holds a third of a cylinder or prism with the same base and height.
Volume and litres
| Cubic unit | Litres |
|---|---|
| 1 dm³ | 1 litre |
| 1 cm³ | 1 ml (one millilitre) |
| 1 000 cm³ | 1 litre |
| 1 m³ | 1 000 litres |
If we work in dm, the dm³ are already litres. If we work in cm, we divide by 1 000.
Example
Example 1: how many litres does an aquarium hold?
The aquarium is 50 cm long, 30 cm wide and 40 cm high. We want to know how many litres it holds.
The base area is 1 500 cm². A layer that is 1 cm high then holds 1 500 cm³.
50 · 30 = 1 500 cm²
The aquarium is 40 cm high, so we stack 40 layers.
V = 1 500 · 40 = 60 000 cm³
1 000 cm³ is one litre, so we divide by 1 000.
60 000 ÷ 1 000 = 60 litres
In dm: 5 · 3 · 4 = 60 dm³, and 60 dm³ is 60 litres.
Example 2: a tin shaped like a cylinder
The tin has radius 4 cm and height 12 cm. We want to find the volume: base area · height.
The base is a circle: r² = 4 · 4 = 16, times π ≈ 3.14.
B ≈ 3.14 · 16 = 50.24 cm²
The tin is 12 cm high, so we stack 12 layers.
V ≈ 50.24 · 12 = 602.88 cm³
We divide by 1 000 to get litres.
602.88 ÷ 1 000 ≈ 0.6 litres
The tin holds about 6 dl.
A box of 8 · 8 · 12 cm holds the tin exactly and has a volume of 768 cm³. The tin should be smaller, and 602.88 cm³ is.
Pitfalls
Adding the measurements: 5 cm, 4 cm and 3 cm make 12 cm³.
Layers of cubes mean multiplication: 5 · 4 · 3 = 60 cm³.
Thinking that 1 litre is 100 cm³, or that 1 m³ is 100 litres.
1 litre is a cube with side 10 cm: 10 · 10 · 10 = 1 000 cm³. And 1 m³ = 1 000 litres.
Using the diameter instead of the radius in the base of the cylinder.
The area of a circle is π · r². Divide the diameter by two first, otherwise the volume is four times too large.
Practise on your own
0 of 4 correct1A cube has sides of 5 cm. What is the volume?
2A cuboid is 8 cm long, 5 cm wide and 2.5 cm high. What is the volume?
3A box is 20 cm long, 10 cm wide and 15 cm high. How many litres does it hold?
4A cylinder has radius 5 cm and height 20 cm. What is the volume? Use π ≈ 3.14.
Common questions
How many litres are there in a cubic metre?
1 m³ is a cube with side 10 dm: 10 · 10 · 10 = 1 000 dm³. That is 1 000 litres.
What is the difference between volume and area?
Area is a surface in two directions, cm². Volume is a space in three directions, cm³.
Which side is the height of a prism lying on its side?
The distance between the two identical end faces. For a tent, a prism lying on its side, it is the length of the tent.
How many dl is 1 000 cm³?
1 000 cm³ is 1 litre, which is 10 dl. 100 cm³ is 1 dl.
Read also
- Years 4–9Area and perimeterHow do you work out area and perimeter?
- Years 6–9Area and circumference of a circleHow do you work out the area and circumference of a circle?
- Years 4–9Converting unitsHow do you convert units?
- Years 7–9 and upper secondaryPowers and the laws of exponentsHow do you calculate with powers?
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