Volume Calculator
A free volume calculator for the seven most common 3D shapes — cube, rectangular prism, sphere, cylinder, cone, pyramid, and triangular prism. Choose your shape, enter the required dimensions in any consistent unit (cm, m, in, ft), and the calculator shows the formula being applied, the substituted values, and the final volume in cubic units. When you enter dimensions in centimetres, the result is also displayed in liters (1 liter = 1,000 cm³), which is convenient for cooking, container sizing, and aquarium calculations. The standard formulas used: cube V = s³; prism V = l × w × h; sphere V = (4/3)πr³; cylinder V = πr²h; cone V = (1/3)πr²h; pyramid V = (1/3) × base area × h. A useful relationship: a cone has exactly one-third the volume of a cylinder with the same base and height. Use this for math homework, figuring out how much paint or concrete fills a tank, sizing storage containers, or any scenario that needs a 3D measurement. Pick a shape and enter dimensions below.
Formula: V = πr²h
Result
| Volume (cubic units): | 282.74333882 |
| If units = cm, liters: | 0.28274334 L |
| If units = cm, milliliters: | 282.74333882 mL |
V = π × r² × h V = 3.14159265 × 3² × 10 V = 3.14159265 × 9 × 10 V = 282.74333882
Volume Formulas for Common 3D Shapes
| Shape | Formula | Variables |
|---|---|---|
| Cube | V = s³ | s = side length |
| Rectangular prism | V = l × w × h | l = length, w = width, h = height |
| Sphere | V = (4/3)πr³ | r = radius |
| Cylinder | V = πr²h | r = base radius, h = height |
| Cone | V = (1/3)πr²h | r = base radius, h = height |
| Square pyramid | V = (1/3) × b² × h | b = base side, h = height |
| Triangular prism | V = (½ × b × h) × l | b, h = triangle base/height, l = prism length |
Understanding the Cylinder Formula
The cylinder volume V = πr²h is simply the area of the circular base (πr²) multiplied by the height h. This is the "prism principle": Volume = base area × height. This applies to any prism with a uniform cross-section — rectangular, triangular, or otherwise.
Example: A cylindrical water tank with radius 50 cm and height 120 cm: V = π × 2500 × 120 = 942,478 cm³ ≈ 942 liters.
The Cone and Pyramid: Why the 1/3 Factor?
A cone has exactly 1/3 the volume of a cylinder with the same base and height. A square pyramid has exactly 1/3 the volume of a rectangular prism with the same base and height. This 1/3 relationship holds for any pyramid or cone and can be proved by Cavalieri's principle or by integration.
Practical implication: filling a conical funnel three times fills one cylindrical can of the same diameter and height. An ice cream cone holds 1/3 as much as a cylindrical container of the same dimensions.
The Sphere Formula Derived
V = (4/3)πr³ comes from integrating circular cross-sections along the sphere's axis. Each cross-section at height y has radius √(r²−y²) and area π(r²−y²). Integrating from −r to r gives (4/3)πr³.
The surface area of a sphere is 4πr² — notably, the derivative of (4/3)πr³ with respect to r, which makes geometric sense: adding a thin shell of thickness dr increases volume by surface area × dr.
Volume Unit Conversions
| From | To | Multiply by |
|---|---|---|
| cm³ (mL) | Liters | 0.001 |
| Liters | cm³ | 1,000 |
| m³ | Liters | 1,000 |
| ft³ | Liters | 28.317 |
| US gallons | Liters | 3.785 |
| UK gallons | Liters | 4.546 |
| Cubic inches | cm³ | 16.387 |
Real-World Volume Examples
- Concrete slab: A 20 ft × 10 ft slab 4 inches thick. Convert to feet: 20 × 10 × (1/3) = 66.67 ft³. Concrete is sold in cubic yards: 66.67 / 27 ≈ 2.47 cubic yards. Our Concrete Calculator handles this directly.
- Water tank: A cylindrical tank 2 m diameter, 3 m tall: r = 1 m, V = π × 1 × 3 ≈ 9.42 m³ = 9,425 liters. At 1 kg/liter, it holds 9.4 metric tonnes of water.
- Aquarium: A rectangular tank 90 cm × 40 cm × 45 cm: V = 90 × 40 × 45 = 162,000 cm³ = 162 liters. Filled 80% (use our Percentage Calculator for quick conversions): 129.6 liters of water needed.
- Basketball: NBA ball diameter 9.4 inches (r = 4.7 in): V = (4/3)π × 4.7³ ≈ 435 in³ ≈ 7.1 liters.
Frequently Asked Questions
What is volume and how is it measured?
Volume is the amount of 3D space enclosed by a solid shape, measured in cubic units (cm³, m³, ft³, etc.). The basic unit relationship: 1 liter = 1,000 cm³ (1 dm³). Volume determines how much liquid a container holds, how much material a solid object contains, and is used in engineering, cooking, medicine, and construction.
What is the volume formula for a sphere?
Volume of a sphere = (4/3)πr³, where r is the radius. A sphere with radius 5 cm has volume = (4/3) × π × 125 ≈ 523.6 cm³ = 0.524 liters. The formula comes from integrating thin circular disks across the sphere's diameter. Surface area of a sphere = 4πr² (different from volume).
How is the volume of a cone related to a cylinder?
A cone with the same base radius and height as a cylinder has exactly one-third the volume: V_cone = (1/3)πr²h vs V_cylinder = πr²h. This means three identical cones fit perfectly inside one cylinder. This 1/3 relationship also holds for pyramids vs. rectangular prisms.
How do I convert cubic centimeters to liters?
1 liter = 1,000 cm³ (cubic centimeters = milliliters). So to convert cm³ to liters, divide by 1000. For example, 523.6 cm³ = 0.5236 liters. Other useful conversions: 1 m³ = 1,000 liters = 1,000,000 cm³. 1 ft³ ≈ 28.317 liters. 1 US gallon = 3,785.4 cm³ = 3.785 liters.
What is the difference between volume and surface area?
Volume measures the 3D space inside a shape (cubic units). Surface area measures the total 2D area of all faces/surfaces of the shape (square units). For a cube with side s: Volume = s³ but Surface area = 6s². For a sphere: Volume = (4/3)πr³ but Surface area = 4πr². Volume grows faster than surface area — doubling dimensions multiplies volume by 8 but surface area only by 4.