Short answer: V = πr²h. Square the radius, times it by the height, and keep π at the front. Writing the volume of a cylinder in terms of pi simply means leaving that π as a symbol, so a can with a 2 cm radius and a 10 cm height holds exactly 40π cm³. Most IGCSE and CBSE papers want the answer in this exact form.

Volume of a cylinder at a glance

Using the radius: V = πr²h

Using the diameter: V = πd²h / 4

Hollow cylinder (pipe): V = π(R² − r²)h

In this guide:

• what volume means for a cylinder, and why the formula works
• the formula using the radius or the diameter
• volume of right, oblique, elliptical and hollow cylinders
• 9 step-by-step examples, with practice questions

What is the volume of a cylinder?

Volume is the amount of 3D space inside a shape. For a container, it is the capacity: how much water, juice or fuel it can hold.

Picture a cylinder as a stack of identical coins. Each coin is a thin circle of area πr². Stack enough of them to reach height h and the total space they fill is the area of one circle times the height of the stack.

Quick check: a cylinder with radius 2 cm and height 10 cm has base area π × 2² = 4π cm², so its volume is 4π × 10 = 40π cm³ (about 125.7 cm³).

"In terms of pi" means you keep π as a symbol instead of replacing it with 3.14 or 22/7. The answer stays exact. Only change it to a decimal when the question asks for one.

Volume of a cylinder formula

V = \pi r^2 h

• r: half the width of the circular end
• h: the straight-up distance between the two circular ends
• V: the space inside, given in cubic units such as cm³ or m³

Using the diameter

If you're given the diameter d, then r = d/2. Substituting gives:

V = \pi \left(\frac{d}{2}\right)^2 h = \frac{\pi d^2 h}{4}

For example, d = 10 cm and h = 4 cm give V = π × 100 × 4 ÷ 4 = 100π cm³.

Why the formula works

1. Slice a cylinder anywhere parallel to its ends and you always see the same circle. Solids like this are measured as cross-section × length.
2. That circular slice covers πr². Our Area of a Circle guide explains where this comes from.
3. Repeat the slice all the way up through height h and you get πr² × h, written V = πr²h.

Rearranging the formula

• Know V and r, need h: share the volume out over the end area, so h = V ÷ πr²
• Know V and h, need r: divide V by πh, then square-root the result: r = √(V ÷ πh)

Volume of different types of cylinders

1. Right circular cylinder. This is the everyday shape of a can or a glass, with walls at 90° to the round ends. Use V = πr²h as it stands.

2. Oblique cylinder. Here the walls lean, like a stack of coins nudged sideways. Leaning the stack moves the coins but doesn't remove any, so the space taken up is unchanged and V = πr²h still works. The one catch: h must be measured straight up between the two ends. The slanted wall length gives the wrong answer.

3. Elliptical cylinder. Picture a cylinder squashed so its ends become ovals, like some tins of fish. An oval with half-widths a and b covers πab, so V = πabh. Make a and b equal and you are back to an ordinary round cylinder.

4. Hollow cylinder (a pipe or tube). Work out the whole solid using the outside radius R, then take away the empty core using the inside radius r. What remains is the material the pipe is made of:

V = \pi R^2 h - \pi r^2 h = \pi (R^2 - r^2) h

Image suggestion: four cylinders side by side (right, oblique, elliptical and hollow), with r, R, a, b and h labelled.

Solved examples

Each example follows the same steps: note what's given, convert the diameter to a radius if needed, substitute into the formula, then simplify.

Example 1: A can has radius 5 cm and height 12 cm. Find its volume in terms of π

• Given: r = 5 cm, h = 12 cm
• Formula: V = πr²h
• Substitute: π × 25 × 12
• Answer: V = 300π cm³ (≈ 942 cm³)

Example 2: A round pool is 8 m across and 2.5 m deep. How much water fills it?

• Given: d = 8 m, so r = 4 m. h = 2.5 m
• Substitute: π × 16 × 2.5
• Answer: V = 40π m³ (≈ 125.7 m³)

Example 3: A cylinder of volume 360π cm³ has radius 6 cm. How tall is it?

• Given: V = 360π, r = 6
• Rearrange: h = V ÷ πr² = 360π ÷ 36π
• Answer: h = 10 cm

Example 4: A cylinder of volume 576π cm³ is 9 cm tall. Find its radius

• Given: V = 576π, h = 9
• Rearrange: r² = 576π ÷ 9π = 64
• Answer: r = 8 cm

Example 5: A metal pipe is 50 cm long, with outer radius 10 cm and inner radius 9 cm. Find the volume of metal

• Given: R = 10, r = 9, h = 50
• Formula: V = π(R² − r²)h
• Substitute: π(100 − 81) × 50 = π × 19 × 50
• Answer: V = 950π cm³

Example 6: An oval tin has semi-axes 5 cm and 2 cm and is 7 cm tall. Find its volume

• Given: a = 5, b = 2, h = 7
• Formula: V = πabh
• Substitute: π × 5 × 2 × 7
• Answer: V = 70π cm³

Example 7: A leaning cylinder has radius 4 cm, slanted side 12 cm and perpendicular height 9 cm. Find its volume

• Which height? Use the perpendicular height, 9 cm, not the 12 cm slant.
• Substitute: π × 16 × 9
• Answer: V = 144π cm³

Example 8: How many litres does a bottle hold with radius 4 cm and height 20 cm?

• Volume: π × 16 × 20 = 320π ≈ 1,005.3 cm³
• Convert: 1,000 cm³ = 1 litre
• Answer: about 1.01 litres

Example 9: What happens to the volume if the radius doubles?

• Reason: volume depends on r², and (2r)² = 4r²
• Answer: the volume becomes 4 times larger. Doubling only the height just doubles the volume.

Where the volume of a cylinder is used

• Water tanks and pools: how much water they hold
• Cans, bottles and tins: the capacity printed on the label
• Pipes: the water a pipe carries, using the inner radius, or the metal it's made from, using the hollow formula
• Engineering: engine cylinders, where the volume swept by the piston sets the engine size

In the curriculum: cylinder volume appears in IGCSE Maths (Core and Extended) mensuration, IB MYP Maths, and CBSE Class 9–10 surface areas and volumes. IGCSE questions often ask for an answer "in terms of π" or to 3 significant figures.

Common mistakes

1. Using the diameter as the radius. Halve the diameter first.
2. Not squaring the radius. The formula is πr²h, not πrh.
3. Mixing up volume and surface area. Volume (πr²h) is in cubic units. Total surface area, 2πr(r + h), is in square units.
4. Using the slant height of an oblique cylinder. Always use the perpendicular height.
5. Subtracting radii instead of areas for a pipe. It is π(R² − r²)h, not π(R − r)²h.
6. Replacing π with 3.14 when the question asks for an answer in terms of π.
7. Mixing units. Convert everything to cm or m before you multiply.

Practice questions

1. Find the volume of a cylinder with radius 9 cm and height 2 cm, in terms of π.
2. A cylinder has diameter 20 cm and height 3 cm. Find its volume in terms of π.
3. A cylinder of volume 96π cm³ has radius 4 cm. Find its height.
4. A cylinder of volume 500π cm³ is 20 cm tall. Find its radius.
5. A tube has outer radius 5 cm, inner radius 3 cm and length 11 cm. Find the volume of material.
6. An elliptical cylinder has semi-axes 7 cm and 4 cm and height 5 cm. Find its volume.
7. A tank has radius 1.5 m and height 2 m. How many litres does it hold? (1 m³ = 1,000 litres)
8. A cylinder's radius is tripled and its height is halved. By what factor does its volume change?

Answers

1. π × 81 × 2 = 162π cm³
2. r = 10, so π × 100 × 3 = 300π cm³
3. h = 96π ÷ 16π = 6 cm
4. r² = 500π ÷ 20π = 25, so r = 5 cm
5. π(25 − 9) × 11 = 176π cm³
6. π × 7 × 4 × 5 = 140π cm³
7. π × 2.25 × 2 = 4.5π ≈ 14.14 m³, about 14,137 litres
8. 3² × ½ = 4.5 times larger

Frequently asked questions

What is the volume of a cylinder in terms of pi?

It's πr²h with the π left in. A 2 cm radius and 10 cm height, for instance, give 40π cm³ rather than 125.66 cm³.

What is the formula for the volume of a cylinder?

V = πr²h. Work out the round end first, then multiply by how tall the cylinder stands.

How do you find the volume of a cylinder from its diameter?

Either halve the diameter and carry on as normal, or skip that step with the version V = πd²h ÷ 4.

How do you find the height of a cylinder from its volume?

Work out the end area πr², then see how many times it fits into the volume: h = V ÷ πr².

How do you find the radius of a cylinder from its volume?

Share the volume out by π times the height, then square-root what's left: r = √(V ÷ πh).

What is the volume of an oblique cylinder?

It matches an upright cylinder of the same size, πr²h. Just be sure h is measured vertically between the ends.

What is the volume of a hollow cylinder?

Take the full outer cylinder and remove the empty core: π(R² − r²)h. The answer is how much material the tube contains, not how much it can carry.

What is the volume of an elliptical cylinder?

Multiply the oval end, πab, by the height to get πabh. Here a and b are the two half-widths of the oval.

What is the difference between the volume and surface area of a cylinder?

One tells you how much a cylinder holds (πr²h, in cm³), and the other how much label or metal wraps around it (2πr(r + h), in cm²). Mixing them up is one of the most common exam slips.

How do you convert a cylinder's volume to litres?

A litre is 1,000 cm³, so divide an answer in cm³ by 1,000. A cubic metre holds 1,000 litres, so multiply an answer in m³ by 1,000.

Book your free demo class with MentorMatch

Volume questions get tricky when the diameter is given instead of the radius, when the answer must stay in terms of π, or when the cylinder is hollow.

A MentorMatch tutor works one-to-one with your child, builds a clear method for every type of cylinder question, and practises the exam-style problems that appear in IGCSE papers.

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