Learn: how much fits inside?
Imagine a straight-sided circular container. Its inside radius is 3 cm and its inside height is 10 cm. We want the space it can hold, measured in cubic centimetres (cm³). One cubic centimetre is a cube 1 cm wide, 1 cm deep and 1 cm tall.
The radius r goes from the centre of a circular base to its edge. The diameter d goes across the entire circle, so r = d ÷ 2. The height h is the perpendicular distance between the bases. π (pi) is approximately 3.14159; keep the calculator’s π value until the final rounding.
Why V = πr²h?
A circular base has area πr². Imagine equal, very thin circular layers stacked through height h: base area multiplied by height gives volume. Square centimetres × centimetres gives cubic centimetres. For an oblique cylinder the same formula uses perpendicular height, not the sloping side length.
V = πr²h. If diameter is given, V = π(d/2)²h.
Worked example: the container
- Set up: r = 3 cm and h = 10 cm. Both lengths use the same unit.
- Find base area: π × 3² = 9π cm².
- Multiply by height: 9π × 10 = 90π cm³.
- Interpret: approximately 282.74 cm³ of internal space, if the inside is an ideal cylinder. Actual containers may have curved ends or thick walls.
Explore: change one dimension
Predict first: what happens when you double the radius while keeping the height fixed? Compare that with doubling only the height.
V = π × 3² × 10 = 90π ≈ 282.74 cm³
The sliders use whole centimetres. Volume measures the space inside; surface area measures the material covering the outside. The written examples below work without this explorer.
Worked examples with different dimensions
These are idealised geometric models. Use internal dimensions for capacity. The numbers below use π before rounding to two decimals.
| Model and dimensions | Setup and exact answer | Approximate volume |
|---|---|---|
| Water-storage cylinder: r = 8 m, h = 20 m | π × 8² × 20 = 1280π m³ | 4,021.24 m³ |
| Plant container: r = 10 cm, h = 30 cm | π × 10² × 30 = 3000π cm³ | 9,424.78 cm³ |
| Generic cylinder: r = 3 in, h = 25 in | π × 3² × 25 = 225π in³ | 706.86 in³ |
| Generic drum: r = 12 in, h = 34.5 in | π × 12² × 34.5 = 4968π in³ | 15,607.43 in³ |
| Concrete-form cylinder: r = 6 in, h = 48 in | π × 6² × 48 = 1728π in³ | 5,428.67 in³ |
A drum’s geometry does not establish a standard oil-barrel capacity. Likewise, geometric tank volume alone does not tell us the amount of compressed breathing gas or safe diving time.
Mixed units: a garden hose
A straight cylindrical hose interior has radius 1 cm and length 10 m. Convert 10 m to 1,000 cm first. V = π × 1² × 1,000 = 1,000π cm³ ≈ 3,141.59 cm³. Since 1,000 cm³ = 1 litre, that is about 3.14 litres. This is the internal space, not a flow rate.
Practice, then explain your reasoning
1. Diameter 8 cm, height 5 cm: find the volume.
First halve the diameter: r = 4 cm. V = π × 4² × 5 = 80π ≈ 251.33 cm³. Using 8 as the radius would make the answer four times too large.
2. A cylinder has V = 72π cm³ and r = 3 cm. Find its height.
72π = π × 3² × h = 9πh. Divide both sides by 9π to get h = 8 cm. Check: 9π cm² × 8 cm = 72π cm³.
3. Radius doubles, height stays fixed. Why is the new volume four times as large?
The radius is squared: (2r)² = 4r², so V becomes 4πr²h. Doubling height alone makes volume twice as large.
4. Radius 2 cm and height 0.3 m: what must you do before calculating?
Use one length unit. Convert height to 30 cm. V = π × 2² × 30 = 120π ≈ 376.99 cm³.
Review: three checks before you finish
- Did you use radius rather than diameter?
- Did you use perpendicular height and consistent length units?
- Does the answer have cubic units, and did you round only at the end?
Try a new problem tomorrow and explain why the formula fits. Correct answers on this short set alone do not establish mastery.
Keep learning
Compare cylinder volume and surface area or explore geometry topics and tutoring.
Optional reference: OpenStax: volume and surface area. This page provides its own explanations and practice.