Maths
How do you find the volume of a cylinder?
Quick answer
Multiply the area of the circular base by the height: V = πr²h. A cylinder with radius 5 cm and height 12 cm holds about 942 cm³, just under a litre.
In one sentence
The volume of a cylinder is the area of its circle-shaped base multiplied by its height: V = πr²h.
Explained
A cylinder is a 3D shape with two equal, parallel circles at the ends and a curved side joining them, like a tin of beans, a candle or a water pipe. Its volume is how much space it takes up, or how much it can hold. For the general idea, see volume.
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Think of a cylinder as a pile of identical coins. Each coin is a thin circle. The area of one circle is πr² (see area of a circle). Stack them up to height h, and the space they fill is the area of the base times the height:
V = π × r² × h, usually written V = πr²h
This is the same rule as for any prism: volume = area of the base × height. A cuboid is a rectangle times its height; a cylinder is a circle times its height.
Here r is the radius of the circular end (half the diameter) and h is the height (or length) of the cylinder. Because you multiply three lengths, the answer is in cubic units: cm³, m³, mm³.
For drinks, tanks and pools you usually want the answer in litres:
- 1 cm³ = 1 ml, so 1000 cm³ = 1 litre
- 1 m³ = 1000 litres
Notice that the radius is squared, but the height is not. That makes the radius much more powerful. Doubling the height doubles the volume, but doubling the radius makes the volume four times as big. A wide, short pot can hold far more than a thin, tall glass.
Do not confuse volume with surface area. Volume is the space inside (cm³). Surface area is the amount of material needed to make the outside (cm²): two circles plus the curved side, 2πr² + 2πrh.
Examples
| Radius r | Height h | r² | Base area πr² (π ≈ 3.14) | Volume |
|---|---|---|---|---|
| 1 cm | 10 cm | 1 | 3.14 cm² | 31.4 cm³ |
| 3 cm | 5 cm | 9 | 28.26 cm² | 141.3 cm³ |
| 5 cm | 12 cm | 25 | 78.5 cm² | 942 cm³ |
| 10 cm | 20 cm | 100 | 314 cm² | 6280 cm³ = 6.28 litres |
| 2 m | 3 m | 4 | 12.56 m² | 37.68 m³ |
Worked example: a tin has a diameter of 8 cm and a height of 10 cm. The radius is 8 ÷ 2 = 4 cm. Base area = 3.14 × 4² = 3.14 × 16 = 50.24 cm². Volume = 50.24 × 10 = 502.4 cm³, which is about half a litre.
Common mistakes
- Using the diameter instead of the radius. Halve the diameter first. With d = 10 cm, use r = 5 cm, not 10 cm (that gives an answer four times too big).
- Squaring the wrong thing. Only the radius is squared: π × r × r × h. Not (π × r)², and not 2 × r.
- Mixing up the formulas. πr²h is the volume; 2πrh is the curved surface; ⅓πr²h is a cone.
- Wrong units or conversions. Volume is in cm³ or m³. And 1 m³ is 1000 litres, not 100.
Step by step
Step by step
- Find the radiusIf you are given the diameter, halve it. Make sure r and h are in the same unit.
- Square the radiusWork out r × r.
- Multiply by πThis gives the area of the circular base, πr².
- Multiply by the heightV = πr² × h.
- Write cubic unitsCm³ or m³. Convert to litres if asked: ÷ 1000 from cm³, × 1000 from m³.
Working backwards: h = V ÷ (πr²) gives the height, and r = √(V ÷ (πh)) gives the radius. In "terms of π" questions, leave π as a letter: r = 4 cm and h = 7 cm give V = π × 16 × 7 = 112π cm³.
Practice questions
Find the volume (use π ≈ 3.14 unless told otherwise)
Try each one, then check your answer.
- r = 1 cm, h = 5 cm.
Show answer
3.14 × 1 × 5 = 15.7 cm³ - r = 3 cm, h = 10 cm.
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3.14 × 9 × 10 = 282.6 cm³ - r = 5 m, h = 2 m.
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3.14 × 25 × 2 = 157 m³ - Diameter 8 cm, height 5 cm.
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r = 4 cm; 3.14 × 16 × 5 = 251.2 cm³ - Diameter 20 cm, height 15 cm.
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r = 10 cm; 3.14 × 100 × 15 = 4,710 cm³ - r = 4 cm, h = 7 cm. Give the volume in terms of π.
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π × 16 × 7 = 112π cm³ - r = 7 cm, h = 10 cm, using π ≈ 22/7.
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22/7 × 49 × 10 = 22 × 7 × 10 = 1,540 cm³ - r = 0.5 m, h = 2 m.
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3.14 × 0.25 × 2 = 1.57 m³ - Write 4,710 cm³ (question 5) in litres.
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4,710 ÷ 1,000 = 4.71 litres - A cylinder has volume 314 cm³ and radius 5 cm. Find its height.
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Base area = 3.14 × 25 = 78.5 cm²; h = 314 ÷ 78.5 = 4 cm - A cylinder has volume 1,130.4 cm³ and height 10 cm. Find its radius.
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r² = 1,130.4 ÷ (3.14 × 10) = 36, so r = 6 cm - A cylinder has volume 500π cm³ and radius 5 cm. Find its height.
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500π = π × 25 × h, so h = 20 cm - Cylinder A has r = 2 cm, h = 10 cm. Cylinder B has r = 4 cm, h = 5 cm. Which holds more?
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A: π × 4 × 10 = 40π; B: π × 16 × 5 = 80π. B holds twice as much - A pipe is 100 cm long, with outer radius 5 cm and inner radius 4 cm. What volume of material is it made of?
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π × (25 − 16) × 100 = 900π ≈ 2,826 cm³ - A cylinder's height equals its diameter, and its radius is 3 cm. Find its volume.
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h = 6 cm; 3.14 × 9 × 6 = 169.56 cm³
Word problems
Cylinders in real life
Try each one, then check your answer.
- A water tank has radius 0.5 m and height 1.2 m. How many litres does it hold?
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3.14 × 0.25 × 1.2 = 0.942 m³ = 942 litres - A glass has radius 3 cm and height 12 cm. How many full glasses can you pour from a 1.5-litre bottle?
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One glass = 3.14 × 9 × 12 = 339.12 cm³; 1,500 ÷ 339.12 ≈ 4.4, so 4 full glasses - A round paddling pool has diameter 4 m and is filled to a depth of 1 m. How many litres of water is that?
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r = 2 m; 3.14 × 4 × 1 = 12.56 m³ = 12,560 litres - A round cake tin has diameter 20 cm. The batter is 4 cm deep. What volume of batter is in the tin?
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r = 10 cm; 3.14 × 100 × 4 = 1,256 cm³ - A hose is 5 m long with an inside radius of 1 cm. How much water is inside it when full?
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5 m = 500 cm; 3.14 × 1 × 500 = 1,570 cm³ ≈ 1.57 litres - A cylindrical tank holds 6,280 litres and has a radius of 1 m. How tall is it?
Show answer
6,280 litres = 6.28 m³; h = 6.28 ÷ (3.14 × 1) = 2 m
Questions people ask
What is the formula for the volume of a cylinder?
V = πr²h: pi times the radius squared times the height. It is the area of the circular base multiplied by the height.
How do you find the volume of a cylinder with the diameter?
Halve the diameter to get the radius, then use V = πr²h. If you prefer, V = π × (d ÷ 2)² × h = πd²h ÷ 4 gives the same answer.
What is the volume of a cylinder measured in?
In cubic units: cm³, m³, mm³, or cubic inches and cubic feet in the US. For liquids, convert to litres: 1000 cm³ = 1 litre and 1 m³ = 1000 litres.
How do you find the volume of a cylinder in gallons?
Work out the volume in cubic feet first, then multiply by about 7.48 to get US gallons (1 cubic foot ≈ 7.48 US gallons). If you have litres, divide by about 3.785. For cubic yards, divide cubic feet by 27.
How do you measure the volume of a real cylinder?
Measure straight across the circular end through the middle (the diameter) and halve it to get the radius. Measure the height. Then use V = πr²h. For a container, measure the inside to find how much it holds.
Does a slanted cylinder have the same volume?
Yes, as long as you use the perpendicular height (straight up, not along the slant). A leaning stack of coins holds the same number of coins as a straight one.
What is the difference between volume and surface area of a cylinder?
Volume (πr²h, in cm³) is the space inside. Surface area (2πr² + 2πrh, in cm²) is the area of the outside: the label plus the two lids of a tin.
Quick quiz
Volume of a cylinder
Check what you know with 5 quick questions.
Question 1 of 5
Which formula gives the volume of a cylinder with radius r and height h?
Practise it
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For grown-ups
Related topics
Volume
Volume is the amount of space a 3D object takes up, measured in cubic units like cm³ or m³. A box 4 cm by 3 cm by 2 cm has a volume of 24 cm³.
Area of a circle
The area of a circle is pi times the radius squared: A = πr². A circle with radius 5 cm has an area of about 78.5 square centimetres.
Surface area
Surface area is the total area of all the outside faces of a 3D shape. For a cube with side a, it is 6a², because a cube has six equal square faces.
Capacity
Capacity is how much a container can hold, usually measured in millilitres (ml) and litres (l). A large bottle of water often holds 2 litres, which is 2000 ml.
Angles in parallel lines
When a line crosses two parallel lines, corresponding (F) and alternate (Z) angles are equal and co-interior (C) angles add to 180°. Vertically opposite angles are always equal.
Area of a trapezium (trapezoid)
Add the two parallel sides, multiply by the height and halve: A = ½(a + b)h. Parallel sides of 4 cm and 6 cm, 3 cm apart, give an area of 15 cm².