To find the volume of a cylinder, multiply the area of its circular base by its height: V = πr²h. A cylinder with radius 2 and height 5 has base area π × 4 = 12.57, so its volume is 12.57 × 5 = 20π ≈ 62.83 cubic units. Surface area is 28π ≈ 87.96.
A cylinder is a circle stretched upward — so its volume is just the circle’s area, stacked height times. Once you see that, the formula stops being something to memorise and becomes something to build.
Check any value instantly with a free volume calculator — its Cylinder mode shows volume, surface area and the exact multiple of π.
Step 1: The Formula — Base Area Times Height
V = πr²h, where r is the radius of the circular base and h is the height.
The formula has three parts, and each one means something: πr² is the area of the circle at the bottom, and h is how many layers of that circle are stacked. A cylinder is a prism with a round base — and like every prism, its volume is base area × height. That is the whole idea; the rest is arithmetic.
Step 2: Worked Example — r = 2, h = 5
- Square the radius: r² = 2² = 4.
- Find the base area: πr² = π × 4 = 12.57 square units.
- Multiply by the height: V = 12.57 × 5 = 62.83 cubic units.
- Or keep it exact: V = π × 4 × 5 = 20π ≈ 62.83 — the exact answer is 20π, and 62.83 is its decimal.
Surface area alongside it: SA = 2πrh + 2πr² = 2π × 2 × 5 + 2π × 4 = 20π + 8π = 28π ≈ 87.96 square units. The lateral area (the curved side) is 20π = 62.83 — the same number as the volume, which is a coincidence of these particular measurements, not a rule.
Step 3: More Worked Examples
| Cylinder | Base area πr² | Volume V = πr²h | Surface area SA = 2πrh + 2πr² |
|---|---|---|---|
| r = 1, h = 10 | 3.14 | 10π = 31.42 | 22π = 69.12 |
| r = 2, h = 5 | 12.57 | 20π = 62.83 | 28π = 87.96 |
| r = 3, h = 4 | 28.27 | 36π = 113.10 | 42π = 131.95 |
| r = 3, h = 7 | 28.27 | 63π = 197.92 | 60π = 188.50 |
| r = 5, h = 10 | 78.54 | 250π = 785.40 | 150π = 471.24 |
Notice the pattern: doubling the radius quadruples the volume (r is squared), while doubling the height only doubles it. A cylinder with r = 3 and h = 4 has volume 113.10 — the same number as a sphere of radius 3, another coincidence of the measurements (36π both ways). And r = 3, h = 7 is the one row where the surface area number (188.50) is smaller than the volume number (197.92), because a tall thin cylinder holds more than its skin suggests.
Step 4: When You Are Given the Diameter
Problems often give the diameter, not the radius. Halve it first: r = d ÷ 2.
Example: a cylinder with diameter 6 and height 4. Radius r = 6 ÷ 2 = 3, so V = π × 9 × 4 = 36π ≈ 113.10 cubic units, and SA = 2π × 3 × 4 + 2π × 9 = 24π + 18π = 42π ≈ 131.95 square units.
The trap: using the diameter as the radius gives V = π × 36 × 4 = 144π ≈ 452.39 — four times too big, because 2² = 4. On a sphere the same mistake costs eight times; on a cylinder it costs four. Either way, “r = d ÷ 2” is the first line of the working, every time.
Step 5: Volume in Litres — What a Cylinder Holds
Volume in cm³ converts to litres by dividing by 1,000, since 1 litre = 1,000 cm³. A cylinder with r = 2 cm and h = 5 cm holds 62.83 cm³ ≈ 0.06 litres — about a shot glass. A cylinder with r = 5 cm and h = 10 cm holds 785.40 cm³ ≈ 0.79 litres — just under a standard bottle. A cylinder with r = 3 cm and h = 7 cm holds 197.92 cm³ ≈ 0.20 litres. If the question asks “how much water does it hold?”, the answer is the volume in litres, not the volume in cm³ — and never the surface area, which is the amount of material the container is made of, not what fits inside it.
Common Mistakes When Finding Cylinder Volume
Using the diameter as the radius. Diameter 6 means radius 3, not 6. The wrong radius gives 452.39 instead of 113.10 for h = 4 — four times too big.
Forgetting to square the radius. V = π × r × h is not the formula — it is π × r² × h. With r = 2 and h = 5, forgetting the square gives π × 2 × 5 = 31.42, exactly half the right answer of 62.83.
Reporting surface area as volume. The lateral area alone for r = 2, h = 5 is 62.83 — the same digits as the volume. The full surface area is 87.96, and neither is the volume. Write the units: cm³ for volume, cm² for surface area.
Mixing units. Radius in cm and height in metres gives a nonsense answer. Convert first, then calculate: 5 cm is 0.05 m, not 5 m.
Using the slant height. Cylinders have no slant height — that belongs to cones. If a problem gives a diagonal measurement for a cylinder, it is a distractor or a different shape’s number.
Frequently Asked Questions
How do you find the volume of a cylinder?
Multiply the base area by the height: V = πr²h. For r = 2 and h = 5: π × 4 × 5 = 20π ≈ 62.83 cubic units. Square the radius first, multiply by π, then multiply by the height.
What is the formula for the volume of a cylinder?
V = πr²h, where r is the radius of the circular base and h is the height. It is base area (πr²) times height — the same idea as any prism, with a round base instead of a rectangular one.
How do you find the volume of a cylinder with diameter?
Halve the diameter to get the radius first: r = d ÷ 2. A cylinder with diameter 6 and height 4 has r = 3, so V = π × 9 × 4 = 36π ≈ 113.10 cubic units. Using 6 as the radius gives 452.39 — four times too big.
What is the surface area of a cylinder?
SA = 2πrh + 2πr² — the curved side plus two circular bases. For r = 2 and h = 5: 20π + 8π = 28π ≈ 87.96 square units. The lateral area alone is 62.83; forgetting the two bases (2 × 12.57) is the classic error.
How many litres does a cylinder hold?
Divide the volume in cm³ by 1,000. A cylinder of 62.83 cm³ holds about 0.06 litres; one of 785.40 cm³ holds about 0.79 litres; one of 197.92 cm³ holds about 0.20 litres.
What is the volume of a cylinder with radius 3 and height 7?
197.92 cubic units. V = π × 9 × 7 = 63π ≈ 197.92. Its surface area is 60π ≈ 188.50 square units — smaller than the volume number, because this cylinder is tall and thin.
Try It Yourself
Enter any radius and height into a free volume calculator and compare with your hand calculation — including r = 2, h = 5 (62.83) and the diameter-6 example (113.10). The Circle Calculator checks the base area πr² on its own (r = 2 → 12.57), and the Triangle Calculator handles the right triangle behind a cone’s slant height when you move on to cones. For the formula in one place, see the cylinder volume formula on the calculator page.