Volume Calculator
Choose a solid, enter its measurements in any length unit, and get its volume as an exact value — in terms of π or a square root where that applies — plus the decimal, the formula with your numbers put in, a labeled figure, and the same volume in cubic units, liters and US gallons. Free, no sign-up; the calculation runs in your browser.
Results
45π cm³
Volume
141.37 cm³
Volume as a decimal
Other measurements
| Measurement | Exact | Decimal |
|---|---|---|
| Base area | 9π cm² | 28.27 cm² |
The working
V = π × r² × h
V = π × r² × h = π × 3² × 5 = 45π
Figure
The same volume in other units
| Unit | Volume |
|---|---|
| mm³ | 141,371.7 |
| cm³ | 141.3717 |
| m³ | 0.0001413717 |
| km³ | 1.4137E-13 |
| in³ | 8.627029 |
| ft³ | 0.004992493 |
| yd³ | 0.0001849072 |
| mi³ | 3.3917E-14 |
| mL | 141.3717 |
| L | 0.1413717 |
| gal | 0.03734644 |
| fl oz | 4.780345 |
How to Use the Volume Calculator
- 1. Choose the solid. Cube, rectangular prism, triangular prism, cylinder, cone, sphere, hemisphere, pyramid with a rectangular base, cone frustum or ellipsoid.
- 2. Enter the measurements. Whole numbers, decimals and fractions such as 3/4 are read exactly. Use one unit for everything, or tick the box to give each measurement its own unit and choose the unit for the answer.
- 3. Read the volume. It appears in exact form first, such as 45π cm³, then as a decimal rounded to the places you choose. Base area, slant height and space diagonal are listed when the solid has them.
- 4. Check the working and convert. The formula is shown with your numbers put in, beside a labeled figure. The last table gives the volume in other units, including liters and US gallons.
Volume Formulas for Common Solids
Volume is the space a solid occupies, in cubic units. Every formula takes the perpendicular height — measured at a right angle to the base — not the slanted edge.
| Solid | Volume | What you need |
|---|---|---|
| Cube | V = a³ | Edge length a |
| Rectangular prism | V = l × w × h | Length, width, height |
| Triangular prism | V = B × L | Area B of the triangular end and the prism length L |
| Cylinder | V = π × r² × h | Radius r and height h |
| Cone | V = (1/3) × π × r² × h | Radius r and height h |
| Sphere | V = (4/3) × π × r³ | Radius r |
| Hemisphere | V = (2/3) × π × r³ | Radius r |
| Pyramid | V = (1/3) × l × w × h | Base length l, base width w, height h |
| Cone frustum | V = (1/3) × π × h × (R² + R × r + r²) | Bottom radius R, top radius r, height h |
| Ellipsoid | V = (4/3) × π × a × b × c | The three semi-axes a, b, c |
The triangular prism's end area comes from Heron's formula when you give the three sides of the triangle, so you never need its height. The cone, pyramid and frustum formulas all contain a one-third: a cone fills exactly one third of the cylinder with the same base and height.
Worked Examples
- Cylinder. Radius 3 and height 5: V = π × 3² × 5 = 45π ≈ 141.37 cubic units.
- Cone. Radius 3 and height 4: V = (1/3) × π × 3² × 4 = 12π ≈ 37.70. The slant height is √(3² + 4²) = 5.
- Sphere. Radius 3: V = (4/3) × π × 27 = 36π ≈ 113.10. The hemisphere with the same radius holds half of that, 18π.
- Cone frustum. Bottom radius 4, top radius 2, height 3: V = (1/3) × π × 3 × (16 + 8 + 4) = 28π ≈ 87.96.
- Pyramid. A 6 by 6 base and height 4: V = (1/3) × 6 × 6 × 4 = 48.
Cubic Units and Capacity
A volume converts with the length factor cubed. One meter is 100 centimeters, so one cubic meter is 100³ = 1,000,000 cubic centimeters. One foot is 12 inches, so one cubic foot is 12³ = 1,728 cubic inches. Capacity units hang off the cubic centimeter: 1 milliliter is exactly 1 cm³, and 1 liter is 1,000 cm³ (so 1 m³ is 1,000 liters). A US gallon is defined as 231 cubic inches, which is 3.785411784 liters.
| Length | Volume |
|---|---|
| 1 m = 100 cm | 1 m³ = 1,000,000 cm³ = 1,000 L |
| 1 ft = 12 in | 1 ft³ = 1,728 in³ ≈ 28.3168 L |
| 1 yd = 3 ft | 1 yd³ = 27 ft³ |
| 1 cm³ | 1 mL |
| 231 in³ | 1 US gallon ≈ 3.7854 L |
For instance, a cylindrical can of radius 10 cm and height 20 cm has volume 2000π cm³ ≈ 6283.19 cm³, which is about 6.28 liters.
Common Mistakes
- Using the slant height of a cone, pyramid or frustum where the formula needs the perpendicular height.
- Entering a diameter where a radius is wanted — the cylinder, cone, sphere and frustum fields take radii.
- Forgetting the one-third in the cone and pyramid formulas.
- Converting cubic units with the plain length factor: 1 m³ is a million cubic centimeters, not a hundred.
Frequently Asked Questions
- How do you calculate the volume of a cylinder?
- Multiply the area of the circular base by the height: V = π × r² × h. A cylinder with radius 3 and height 5 has volume π × 9 × 5 = 45π, about 141.37 cubic units. Use the radius, which is half the diameter.
- Why is the volume of a cone one third of a cylinder's?
- A cone and a cylinder with the same base and height are related by a fixed ratio of 1 to 3: three cones fill the cylinder exactly. Calculus shows this by adding up thin circular slices whose radii shrink linearly to zero, and the shrinking contributes the factor (1/3). The same factor appears in every pyramid.
- How do you find the volume of a sphere?
- Use V = (4/3) × π × r³. Because the radius is cubed, doubling the radius multiplies the volume by eight. A sphere with radius 3 has volume 36π, about 113.10 cubic units, and a hemisphere of that radius has half, 18π.
- What is the volume of a pyramid?
- One third of the base area times the perpendicular height, whatever shape the base is: V = (1/3) × B × h. For a rectangular base that is (1/3) × l × w × h. A pyramid with a 6 by 6 base and height 4 has volume 48.
- How do I convert cubic centimeters to liters?
- Divide by 1,000: 1 liter is exactly 1,000 cm³, and 1 milliliter is exactly 1 cm³. So 6,283.19 cm³ is 6.28 L. For cubic meters multiply by 1,000, since 1 m³ is 1,000 liters.
- What is a frustum, and how do you find its volume?
- A frustum is a cone or pyramid with the top cut off by a plane parallel to the base. For a cone frustum with bottom radius R, top radius r and height h, V = (1/3) × π × h × (R² + R × r + r²). Setting r equal to R turns it into a cylinder, and r = 0 gives a full cone.
- Is the ellipsoid volume formula exact?
- Yes. V = (4/3) × π × a × b × c is exact for every ellipsoid, and it reduces to the sphere formula when the three semi-axes are equal. It is the surface area of a general ellipsoid that has no closed form.