Volume Calculator
Select a 3D shape, enter its dimensions, and get the volume with step-by-step work. Results include both cubic units and the liter equivalent (assuming dimensions in cm).
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.
- 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%: 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.
Related Calculators
- Area Calculator — 2D shape areas: square, rectangle, circle, and more
- Circle Calculator — radius, circumference, area from any input
- Concrete Calculator — cubic yards needed for slabs and footings