Volume in simple words
Volume is how much space a solid thing takes up, or how much it can hold. Think of filling a box with little cubes: the volume is how many cubes fit inside. That is why volume is measured in cubic units, such as cubic meters (m³), or in liters and gallons.
For a box you multiply length × width × height. Many other solids are built from that idea. A cylinder is a stack of circles, and a cone or pyramid holds one third of the matching cylinder or box.
Example: a box is 5 m long, 3 m wide and 4 m tall. Its volume is 5 × 3 × 4 = 60 cubic meters.
Worked example for every solid
Each row substitutes sample dimensions (in meters) into the formula the calculator uses, so you can check a result by hand. Use the Solve for option above to work backward from a known volume, perimeter, diagonal or surface area.
| Solid | Inputs | Formula with numbers | Result |
|---|---|---|---|
| Cube | Side 2 m | V = 2³; S = 6 × 2² | 8 m³; S = 24 m² |
| Rectangular prism | Length 5 m, Width 3 m, Height 2 m | V = 5 × 3 × 2; S = 2 × (5×3 + 5×2 + 3×2) | 30 m³; S = 62 m² |
| Sphere | Radius 3 m | V = 4 × π × 3³ ÷ 3; S = 4 × π × 3² | 113.09734 m³; S = 113.09734 m² |
| Hemisphere | Radius 3 m | V = 2 × π × 3³ ÷ 3; S = 3 × π × 3² | 56.548668 m³; S = 84.823002 m² |
| Cylinder | Radius 2 m, Height 5 m | V = π × 2² × 5; S = 2 × π × 2 × (2 + 5) | 62.831853 m³; S = 87.964594 m² |
| Cone | Radius 3 m, Perpendicular height 4 m | V = π × 3² × 4 ÷ 3; slant l = √(3² + 4²) | 37.699112 m³; S = 75.398224 m² |
| Conical frustum | Bottom radius 4 m, Top radius 2 m, Perpendicular height 3 m | V = π × 3 × (4² + 4×2 + 2²) ÷ 3 | 87.964594 m³; S = 130.79489 m² |
| Square pyramid frustum | Bottom base side 4 m, Top base side 2 m, Perpendicular height 3 m | V = 3 × (4² + 4×2 + 2²) ÷ 3 | 28 m³; S = 57.947332 m² |
| Square pyramid | Base side 6 m, Perpendicular height 4 m | V = 6² × 4 ÷ 3 | 48 m³; S = 96 m² |
| Rectangular pyramid | Base length 6 m, Base width 4 m, Perpendicular height 3 m | V = 6 × 4 × 3 ÷ 3 | 24 m³; S = 62.60387 m² |
| Ellipsoid | Semi-axis a 3 m, Semi-axis b 2 m, Semi-axis c 1 m | V = 4 × π × 3 × 2 × 1 ÷ 3 | 25.132741 m³ |
| Capsule | Radius 1 m, Straight cylinder length 3 m | V = π × 1² × (3 + 4 × 1 ÷ 3); S = 2 × π × 1 × (2 × 1 + 3) | 13.613568 m³; S = 31.415927 m² |
| Hollow cylinder | Outer radius 5 m, Inner radius 4 m, Height 2 m | V = π × (5² − 4²) × 2 | 56.548668 m³; S = 169.646 m² |
| Triangular prism — three triangle sides | Triangle side a 3 m, Triangle side b 4 m, Triangle side c 5 m, Prism length 10 m | s = 6; T = √(6 × 3 × 2 × 1); V = T × 10 | 60 m³; S = 132 m² |
| Hollow cylinder — outer radius and wall thickness | Outer radius 5 m, Wall thickness 1 m, Height 2 m | r = 5 − 1 = 4; V = π × (5² − 4²) × 2 | 56.548668 m³; S = 169.646 m² |
| Triangular prism | Triangle base 6 m, Triangle perpendicular height 4 m, Prism length 10 m | V = 6 × 4 × 10 ÷ 2 | 120 m³ |
Formula and method
Volumes use cubic measures. A rectangular prism is lwh, a cylinder is πr²h, a cone is one-third of a cylinder with the same base and height, and a sphere is 4πr³/3. The displayed shape formula changes with the selected solid.
Worked example
A 5 m by 3 m by 4 m box occupies 60 m³ or 60,000 liters. Its total surface area is 2(15+20+12) = 94 m².
Inputs and limitations
Radii are half diameters; ellipsoid inputs are semi-axes. Capsule height is the straight cylinder section, excluding its curved ends. Surface area includes bases and caps. An arbitrary triangular prism’s base and height do not determine its side lengths, so this tool does not invent a surface area.