Volume Calculator
Calculate the volume of 8 common 3D shapes. Select a shape and enter the dimensions.
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Volume Formulas for Common Shapes
The volume of a 3D shape is the amount of three-dimensional space it occupies, measured in cubic units. Each shape has its own formula based on its geometry.
Cube and Rectangular Box
A cube has all sides equal: V = s³. A rectangular box (cuboid) multiplies all three distinct dimensions: V = length × width × height. These are the simplest volume calculations and form the basis for understanding more complex shapes.
Sphere
The sphere volume formula V = (4/3)πr³ comes from integrating circular cross-sections from pole to pole. A sphere with radius 1 has volume approximately 4.189. Doubling the radius multiplies volume by 8 (2³), showing how volume scales with the cube of linear dimensions.
Cylinder and Cone
A cylinder is a circle extruded along its height: V = πr²h. A cone is a pyramid with a circular base: V = (1/3)πr²h — exactly one-third of the cylinder with the same base and height. This relationship was first proved by Archimedes around 250 BCE.
Pyramid and Prism
All pyramids follow V = (1/3) × base area × height. For a square pyramid: V = (1/3)s²h. A triangular prism has V = (1/2) × base × height_of_triangle × length. Prisms in general equal base area × depth.
Torus
A torus (donut shape) is defined by a major radius R (center to tube center) and minor radius r (tube radius): V = 2π²Rr². This formula comes from Pappus's centroid theorem, multiplying the circle area πr² by the distance traveled by its centroid 2πR.
Volume Formula Quick Reference
| Shape | Formula | Example (common dimensions) |
|---|---|---|
| Cube (s=5) | s³ | 125 cubic units |
| Sphere (r=5) | (4/3)πr³ | ≈523.6 cubic units |
| Cylinder (r=3, h=10) | πr²h | ≈282.7 cubic units |
| Cone (r=3, h=8) | (1/3)πr²h | ≈75.4 cubic units |
| Square Pyramid (s=4, h=6) | (1/3)s²h | 32 cubic units |
Frequently Asked Questions
V = (4/3)πr³ = (4/3)×π×125 ≈ 523.6 cubic units.
V = π × r² × h. Multiply pi by the radius squared, then by the height. For a cylinder with r=3 and h=10: V = π×9×10 ≈ 282.7 cubic units.
V = s³ = 4³ = 64 cubic units. Each side cubed gives the volume.
Volume uses cubic units: cm³, m³, in³, ft³. Common practical units include liters (1 L = 1000 cm³) and gallons (1 US gallon = 231 in³).
A cone's volume is exactly one-third of a cylinder with the same base radius and height: V_cone = (1/3)πr²h = (1/3) × V_cylinder.
1 US gallon = 231 cubic inches. Divide your cubic inch volume by 231 to get gallons. Example: 462 in³ ÷ 231 = 2 gallons.
The 4/3 comes from integrating circular cross-sections over the sphere. Each cross-section at height h has area π(r²−h²), and integrating from −r to +r gives (4/3)πr³. Archimedes first proved this by showing that a sphere fits inside a cylinder with 2/3 of its volume.
Volume scales with the cube of the linear scale factor. Doubling all dimensions multiplies volume by 2³ = 8. A box that is 2×3×4 = 24 cubic units becomes 4×6×8 = 192 cubic units when doubled — eight times larger. This is why scaling up containers has a dramatic effect on capacity.