Volume Calculator (3D Solids, with Diagram)
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A volume calculator finds how much space a solid fills, in cubic units. For a cylinder, V = πr²h: a can with radius 4 cm and height 9 cm holds π × 4² × 9 = 144π ≈ 452.39 cm³. Since 1 liter is 1,000 cm³, that is about 0.45 liters. The tool below covers 13 solids with exact π answers, liters, gallons and working backward.
A cylinder with radius 4 cm and height 9 cm has a volume of 144π cm³ (about 452.39), which is 0.452389 liters.
Solid lines are visible edges, dashed lines are hidden edges or measurements.
Show steps
- Base area = πr² = π × (4)² = 16π ≈ 50.27 cm².
- Volume = base area × height = πr²h = 16π × 9 = 144π ≈ 452.39 cm³.
- A cylinder is a stack of circles, so its volume is always base area × height, like a prism.
- Capacity: 1 liter = 1,000 cm³, so 452.39 cm³ ÷ 1,000 = 0.452389 L (and 0.119509 US gallons).
Convert the volume to other units
Cubic units convert by the cube of the length factor: 1 m = 100 cm, so 1 m³ = 100³ = 1,000,000 cm³. 1 liter = 1,000 cm³ exactly; 1 US gallon = 231 in³ = 3.785411784 L exactly.
How to use the volume calculator
- Pick the solid: cube, box, cylinder, sphere, hemisphere, cone, pyramid, triangular prism, frustum, capsule, spherical cap, ellipsoid or tube.
- Choose the length unit (mm, cm, m, in, ft or yd) and type the measurements. Fractions such as 3/4 and decimals both work.
- To work backward, set Find to a height, radius, edge or length, then enter the volume, in cubic units, liters, gallons or cubic feet.
- Read the volume with its exact π form, plus liters and US gallons. Open Show steps for the working and Convert the volume for 12 other units.
Volume formulas
| Solid | Volume formula | Example |
|---|---|---|
| Cube | V = a³ | a = 3: V = 27 |
| Rectangular prism (box) | V = l × w × h | 4 × 3 × 2 = 24 |
| Cylinder | V = πr²h | r = 4, h = 9: 144π ≈ 452.39 |
| Sphere | V = 4/3 πr³ | r = 3: 36π ≈ 113.10 |
| Hemisphere | V = 2/3 πr³ | r = 3: 18π ≈ 56.55 |
| Cone | V = 1/3 πr²h | r = 3, h = 4: 12π ≈ 37.70 |
| Pyramid | V = 1/3 × base area × h | 6 × 6 base, h = 4: 48 |
| Triangular prism | V = ½ × b × t × L | b = 6, t = 4, L = 10: 120 |
| Conical frustum | V = πh(R² + Rr + r²)/3 | r = 3, R = 6, h = 4: 84π ≈ 263.89 |
| Capsule | V = πr²h + 4/3 πr³ | r = 2, h = 5: 92π/3 ≈ 96.34 |
| Spherical cap | V = πh²(3R − h)/3 | R = 5, h = 2: 52π/3 ≈ 54.45 |
| Ellipsoid | V = 4/3 πabc | 6, 4, 3: 96π ≈ 301.59 |
| Tube (pipe) | V = π(R² − r²)h | R = 5, r = 4, h = 20: 180π ≈ 565.49 |
The radius r is half the diameter. Every prism and cylinder follows one rule, V = base area × height, and every cone and pyramid holds exactly one third of the prism or cylinder with the same base and height. A sphere fills two thirds of the cylinder that just fits around it.
Volume examples
A room: 12 ft × 10 ft × 8 ft
V = 12 × 10 × 8 = 960 ft³. One cubic foot is 1,728 in³ and a US gallon is 231 in³, so 960 ft³ is about 7,181.3 gallons of air.
A ball with diameter 24 cm
The radius is 24 ÷ 2 = 12 cm, so V = 4/3 × π × 12³ = 4/3 × π × 1,728 = 2,304π ≈ 7,238.23 cm³, or about 7.24 liters. Using the diameter in place of the radius would make the answer 8 times too large.
A round pool 12 ft across and 4 ft deep
The radius is 6 ft, so V = π × 6² × 4 = 144π ≈ 452.39 ft³. One cubic foot holds about 7.48052 US gallons, so the pool holds about 3,384.1 gallons when filled to the top.
Working backward: how tall must a 500-liter tank be?
A cylindrical tank has radius 40 cm and must hold 500 L = 500,000 cm³. From V = πr²h, h = V ÷ (πr²) = 500,000 ÷ (1,600π) = 625/(2π) ≈ 99.47 cm. The calculator does this when you set Find to “height” and the volume unit to liters.
A square pyramid 230 m wide and 146 m tall
V = 1/3 × 230 × 230 × 146 = 7,723,400 ÷ 3 ≈ 2,574,466.67 m³. A pyramid this size would hold the same as a box 230 m × 230 m × about 48.67 m.
Volume units: cubic units, liters and gallons
Volume is measured in cubic units because three lengths are multiplied. Changing the length unit changes the volume by the cube of the factor: 1 m = 100 cm, so 1 m³ = 100 × 100 × 100 = 1,000,000 cm³. Liters and gallons measure capacity, the amount a container holds.
| Unit | Equals |
|---|---|
| 1 liter (L) | 1,000 cm³ = 1,000 mL (exactly) |
| 1 cubic meter (m³) | 1,000 L |
| 1 US gallon | 231 in³ = 3.785411784 L (exactly) |
| 1 imperial gallon | 4.54609 L (exactly) |
| 1 cubic foot (ft³) | 1,728 in³ ≈ 7.48052 US gallons ≈ 28.3168 L |
| 1 cubic yard (yd³) | 27 ft³ ≈ 764.555 L |
Volume vs surface area
Volume measures what fits inside a solid (cubic units); surface area measures the outside skin (square units). A box 4 × 3 × 2 has volume 24 but surface area 52. This page shows the surface area only as a quick reference; for a face-by-face breakdown, open edges and nets, use the surface area calculator.
Common volume mistakes
- Using the diameter as the radius. Halve the diameter first; otherwise a sphere comes out 8 times too big and a cylinder 4 times.
- Forgetting the 1/3 for cones and pyramids.
- Using the slant height instead of the vertical height in a cone or pyramid formula.
- Mixing units, such as a radius in inches and a height in feet. Convert everything to one unit first.
- Converting cubic units with the length factor. 1 ft = 12 in, but 1 ft³ = 1,728 in³, not 12 in³.
- Rounding π too early. Keep π in the answer (144π) and round once at the end.
Before you trust an answer, estimate it. A cylinder with r = 4 and h = 9 fits inside an 8 × 8 × 9 box of volume 576, so 452 is reasonable; 4,524 would not be. Work the formula by hand for homework, then use the calculator to check the arithmetic and the units.
Sources
Exact unit definitions checked November 2026:
- NIST Handbook 44 (2026), Appendix C: General Tables of Units of Measurement (gallon = 231 in³, liter = 1 dm³, cubic foot = 1,728 in³)
- NIST Special Publication 811, Appendix B.9 (imperial gallon = 4.54609 L)
Volume practice quiz: 10 questions
Test yourself with 10 multiple-choice questions. Choose an answer and press Submit to see if you are right and why. Your score appears at the end.
V = l × w × h = 5 × 4 × 3 = 60 cm³. Adding the edges gives 12, and 94 cm² is the surface area, not the volume.
V = πr²h = π × 3² × 5 = 45π cm³ ≈ 141.37 cm³. Forgetting to square the radius gives 15π, and using the diameter 6 gives 180π.
The radius is 6 ÷ 2 = 3 in, so V = 4/3 π × 3³ = 36π ≈ 113.10 in³. Using the diameter as the radius gives 288π ≈ 904.78 in³, eight times too much.
A cone holds exactly one third of the cylinder with the same base and height: 30 ÷ 3 = 10 L.
V = 1/3 × base area × h = 1/3 × 36 × 10 = 120 m³. Leaving out the 1/3 gives 360 m³, the volume of the matching prism.
1 ft = 12 in, so 1 ft³ = 12 × 12 × 12 = 1,728 in³. 144 is the number of square inches in a square foot.
From V = πr²h, h = V ÷ (πr²) = 24π ÷ (4π) = 6 ft. Dividing by 2π instead of 4π gives 12 ft.
V = a³, so a = ∛343 = 7 cm, because 7 × 7 × 7 = 343. Dividing by 3 (114.33) or taking the square root (18.52) are common mistakes.
V = 50 × 30 × 40 = 60,000 cm³ = 60 L. One US gallon is 3.785411784 L, so 60 ÷ 3.785411784 ≈ 15.85 gallons. 13.2 would be imperial gallons.
Quiz complete
Frequently asked questions
How do you calculate the volume of a cylinder?
Multiply the area of the circular base by the height: V = πr²h. For r = 3 cm and h = 10 cm, V = π × 9 × 10 = 90π ≈ 282.74 cm³.
How do you calculate volume in cubic units?
Measure every length in the same unit, use the formula for the solid, and write the answer in that unit cubed. A box 2 m × 3 m × 4 m has a volume of 24 m³: 24 cubes, each 1 m on a side, fit inside.
Is volume calculated in liters?
Volume can be written in liters or in cubic units; they measure the same thing. One liter is exactly 1,000 cm³, so a solid measured in centimeters can be changed to liters at the end.
How do you convert volume to liters?
Divide cubic centimeters by 1,000, multiply cubic meters by 1,000, or multiply cubic feet by about 28.3168. For example, 452.39 cm³ ÷ 1,000 ≈ 0.45 L.
How do you find volume in gallons?
Divide cubic inches by 231, or multiply cubic feet by about 7.48052. A tank of 10 ft³ holds about 74.8 US gallons. For imperial gallons, divide liters by 4.54609.
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