The volume of a cylinder is V = πr²h: pi times the radius squared times the height. You find the area of the circular base (πr²), then multiply it by how tall the cylinder is. Use the same unit for both measurements, and the answer comes out in cubic units: cubic inches, cubic feet, cubic centimeters.
This guide shows what each letter means, what to do when you only have a diameter or a circumference, and how to turn the result into cubic feet or cubic yards for a real project. It also includes six worked examples you can copy, and the mistakes that cause most wrong answers.
TL;DR
- The formula is V = πr²h, where r is the radius of the circular base and h is the height.
- If you have the diameter, halve it first. Using the diameter as the radius makes your answer four times too big.
- Keep every measurement in one unit before you multiply. Convert inches to feet (or the reverse) first.
- To go backward, use h = V ÷ (πr²) for height, or r = √(V ÷ πh) for radius.
- For concrete tubes, tanks, pipes and similar jobs, round up when you buy material.
Table of Contents
- The cylinder volume formula
- How to calculate cylinder volume, step by step
- Six worked examples
- When you have the diameter, not the radius
- Converting to cubic feet, gallons and other units
- Special cases you will run into
- Common mistakes and how to avoid them
- Perspective
- Using cylinder volume for DIY projects
- Calculate it faster
- FAQ
- Sources
The cylinder volume formula
The standard statement is the one you will find in any geometry text: the volume of a right cylinder is V = πr²h. The National Park Service uses the same expression in its education page on volume, with r as the radius and h as the height.
In plain terms:
- V is the volume, the amount of three-dimensional space inside the cylinder.
- π (pi) is the constant 3.14159..., the ratio of a circle's circumference to its diameter.
- r is the radius, the distance from the center of the circular base to its edge.
- h is the height, the perpendicular distance between the two circular faces.
The radius and height meanings are the same in every textbook: r belongs to the circular cross-section, and h is how far that circle is stretched.
Why the formula works
Think of a cylinder as a stack of identical, very thin circles. Each circle has an area of πr². Stack enough of them to reach height h and the total space is the base area times the height. That is the same idea as the prism rule V = Ah, and BBC Bitesize derives the cylinder formula exactly this way: the base area is A = πr², so V = πr²h.
This is also why you cannot fill a cylinder neatly with unit cubes. Cubes do not fit a curved shape, so you use the formula instead of counting blocks.
Which value of π should you use?
It depends on what the answer is for.
- 3.14 is fine for homework that tells you to use it, and for quick estimates.
- Your calculator's π button gives the most accurate result and is the better habit for anything you will buy or build from.
- Leaving π in the answer (for example 144π) gives the exact value, which many math classes prefer.
The gap is small but real. For a cylinder with radius 2 ft and height 6 ft, the exact volume is 24π ≈ 75.40 ft³, while 3.14 gives 75.36 ft³. That difference does not matter for a flower pot. On a large tank or a long concrete pour, use the full π.
How to calculate cylinder volume, step by step
- Measure the diameter or the radius. The diameter is the full width across the circle through its center. The radius is half of that.
- Convert to a single unit. If the width is in inches and the height is in feet, convert one of them so both match.
- Square the radius. Multiply r by itself.
- Multiply by π. This gives the base area.
- Multiply by the height. This gives the volume.
- Label the answer in cubic units. Inches become in³, feet become ft³.
Six worked examples
Example 1: radius and height given
A cylinder has a radius of 4 inches and a height of 9 inches.
- Base area: π × 4² = 16π ≈ 50.27 in²
- Volume: 50.27 × 9 ≈ 452.39 in³ (exactly 144π in³)
A similar textbook case with centimeters appears in OpenStax, which works a can with r = 4 and h = 10 to get about 502.7 cm³.
Example 2: diameter given
A round planter is 12 inches across and 8 inches tall.
- Radius: 12 ÷ 2 = 6 in
- Base area: π × 6² = 36π ≈ 113.10 in²
- Volume: 113.10 × 8 ≈ 904.78 in³ (exactly 288π in³)
With 3.14 for π you get 904.32 in³, the figure shown in this CK-12 diameter example. Both are correct for their stated value of π.
Example 3: dimensions in mixed units (a concrete tube form)
You are pouring concrete into a cardboard tube form that is 10 inches in diameter and 4 feet tall. The units do not match, so convert first.
- Height: 4 ft × 12 = 48 in
- Radius: 10 ÷ 2 = 5 in
- Volume: π × 5² × 48 = 1,200π ≈ 3,769.91 in³
Now convert to cubic feet. One cubic foot is 12 × 12 × 12 = 1,728 cubic inches, so:
- 3,769.91 ÷ 1,728 ≈ 2.18 ft³
To express that in cubic yards, divide by 27 (because 3 × 3 × 3 = 27 cubic feet per cubic yard): 2.18 ÷ 27 ≈ 0.081 yd³. If you are pouring several footings, multiply by the number of tubes, then add a margin for spillage. Our concrete calculator does this conversion and shows bags and cubic yards for you.
Example 4: a tank in feet
A vertical water tank has a radius of 2 feet and is 6 feet tall.
- Base area: π × 2² = 4π ≈ 12.57 ft²
- Volume: 12.57 × 6 ≈ 75.40 ft³
Because the inputs were already in feet, the answer is already in cubic feet. No further conversion is needed.
Example 5: finding the height from a known volume
Sometimes you know how much a container must hold and need to size it. Rearrange the formula: h = V ÷ (πr²).
Suppose a canister must hold 100 in³ and its radius is 2 in.
- Base area: π × 2² = 4π ≈ 12.57 in²
- Height: 100 ÷ 12.57 ≈ 7.96 in
The same rearrangement for radius is r = √(V ÷ πh). That last step is a square root, which our square root calculator can handle.
Example 6: only the circumference is known
You wrap a tape around a pipe or a post and read a circumference of 18.85 inches. The circumference of a circle is C = 2πr, so r = C ÷ 2π.
- Radius: 18.85 ÷ 6.283 ≈ 3.00 in
- If the post is 10 in tall: π × 3² × 10 = 90π ≈ 282.74 in³
This is the easiest way to measure a tree trunk, a column or a drum when you cannot get a tape across its middle.
When you have the diameter, not the radius
This is the most common source of wrong answers. The formula needs the radius. If the problem or your tape measure gives the diameter, divide it by 2 before using the formula.
The error is large because the radius is squared. Put a diameter of 12 in the formula by mistake and you square 12 instead of 6: 144 instead of 36, which is four times too big. A quick sanity check: the volume of a cylinder should be a bit less than the volume of a square box that is the same width and height. For Example 2, that box would be 12 × 12 × 8 = 1,152 in³, and the cylinder's 904.78 in³ is about 79% of it, which is correct (π ÷ 4 is about 0.785).
You can also skip the halving by using the diameter version of the formula: V = π × (d ÷ 2)² × h, or equivalently V = πd²h ÷ 4.
Converting to cubic feet, gallons and other units
Work in the unit you want from the start whenever you can. If you need cubic feet, measure or convert the radius and height to feet first, as in Example 4.
If you already have an answer in another unit, convert the volume itself. These steps are plain arithmetic:
- Cubic inches to cubic feet: divide by 1,728
- Cubic feet to cubic yards: divide by 27
- Cubic centimeters to liters: divide by 1,000
Do not divide by 12 when converting cubic inches to cubic feet. Volume is three-dimensional, so the factor is cubed. For other pairs, such as metric to US customary, use the unit converter rather than typing conversion factors from memory.
Special cases you will run into
A cylinder that is partly full
For a cylinder standing upright, the liquid forms a smaller cylinder with the same radius. Replace h with the depth of the liquid. A tank 6 ft tall and filled to 4 ft with a radius of 2 ft holds π × 2² × 4 ≈ 50.27 ft³.
A cylinder lying on its side is different. The liquid cross-section is a circle segment, so the simple formula does not apply to a partial fill. A full horizontal cylinder still uses V = πr²h, with h as its length.
A hollow cylinder (pipe or tube)
Calculate the outer cylinder, then subtract the inner one. That gives V = π(R² − r²)h, where R is the outer radius and r the inner radius.
For a pipe with an outer radius of 3 in, an inner radius of 2.5 in and a length of 20 in:
- π × (9 − 6.25) × 20 = 55π ≈ 172.79 in³ of wall material
- The space inside, available for liquid, is π × 2.5² × 20 ≈ 392.70 in³
A slanted (oblique) cylinder
If the sides lean, the height is still the perpendicular distance between the two bases, not the length of the slanted side. Measure straight up, and the same formula holds.
Percent change in volume
If you want to know how much more a wider container holds, calculate each volume and compare them. Our percentage calculator handles the percent difference. Remember that doubling the radius quadruples the volume, while doubling the height only doubles it.
Common mistakes and how to avoid them
| Mistake | What happens | Fix |
|---|---|---|
| Using diameter as radius | Answer is 4 times too large | Halve the diameter first |
| Mixing inches and feet | Answer is wrong by a large factor | Convert everything to one unit |
| Forgetting to square r | Answer is far too small | Compute r × r before multiplying by π |
| Squaring the whole product | Answer is wildly off | Only r is squared: π × (r²) × h |
| Writing square units | Wrong unit label | Volume uses cubic units (in³, ft³) |
| Rounding π to 3 | Up to 4.5% error | Use 3.14 or the calculator's π key |
Perspective
We build every calculator on Countora around the idea that you should be able to see the method, not just the answer. A cylinder is a good example: the formula is short, but the places people go wrong are the diameter-versus-radius step, the unit mismatch and the rounding.
That is why each of our calculators shows its formula and a worked example, and why we cite the sources behind technical data. In our construction tools we round material amounts up so the result is purchase-ready. In our math tools we show both the exact and the rounded answer, so a student can check their own work against either one.
If you take one habit from this article, make it this: write the unit next to every number before you multiply. Most cylinder errors disappear once the units are visible. You can read how we handle rounding and sources on our methodology page.
BABATOUNDE
Using cylinder volume for DIY projects
The formula turns up in more home projects than most people expect:
- Concrete footings and post holes. A round hole is a cylinder. Multiply the volume by the number of holes, then convert to cubic yards.
- Deck and porch piers. Cardboard forms come in set diameters, so you only need the height.
- Water barrels and rain tanks. Measure the inside diameter and the water depth to find how much you hold.
- Planters and fill soil. Volume tells you how much potting mix to buy.
- Gravel and sand for round beds. Treat the bed as a short cylinder: π × r² × depth.
In every case, round up when you order. Material is sold in whole bags, loads or cubic yards, and a small allowance for uneven ground or spillage keeps you from a second trip.
Calculate it faster
If you do this often, a calculator that shows its work saves time and catches unit mistakes. Countora has free calculators for construction and DIY materials, including a concrete calculator for round and rectangular pours, and a set of math calculators for checking homework steps. Each one shows the formula it uses and a worked example, so you can follow the method rather than trust a number.
For a project estimate, start with the construction tools. For schoolwork, use the math tools to confirm your own working, then redo the problem by hand to practice.
FAQ
What is the formula for the volume of a cylinder?
The formula is V = πr²h. Square the radius, multiply by π, then multiply by the height. The answer is in cubic units, such as cubic inches or cubic feet, matching the unit you measured in.
How do I find the volume of a cylinder from its diameter?
Divide the diameter by 2 to get the radius, then use V = πr²h as usual. A 12 inch diameter and 8 inch height give a radius of 6 inches and a volume of about 904.78 in³. You can also use V = π(d ÷ 2)²h in one step.
How do I calculate the cubic feet in a cylinder?
Measure the radius and height in feet, then apply V = πr²h. A cylinder with a 2 ft radius and 6 ft height holds about 75.40 ft³. If you measured in inches, find the volume in cubic inches and divide by 1,728.
Does the formula change if the cylinder is tilted?
No. The formula stays the same as long as h is the perpendicular height between the two circular bases. Do not use the length of the slanted side.
How do I find the height if I know the volume?
Rearrange the formula to h = V ÷ (πr²). For a volume of 100 in³ and a radius of 2 in, the height is about 7.96 in. Use the same unit for volume and radius, in cubic and linear form.
Sources
- OpenStax, Contemporary Mathematics: Volume and Surface Area
- National Park Service, Math at Saguaro: Volume
- BBC Bitesize, Surface area and volume of a cylinder (KS3 Maths)
- BBC Bitesize, Volume of a cylinder
- K12 LibreTexts, Volume of Cylinders
- CK-12 Foundation, Volume of Cylinders