Volume Calculator
Calculate the volume of common 3D shapes: cube, rectangular box, sphere, cylinder, cone, pyramid, and ellipsoid.
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How to use this calculator
Cube: s³. Box: lwh. Sphere: (4/3)πr³. Cylinder: πr²h. Cone: (1/3)πr²h. Pyramid: (1/3)b²h. Ellipsoid: (4/3)πabc.
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Select the 3D shape from the dropdown.
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Enter the required dimensions (radius, height, side length, etc.).
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The volume is calculated immediately in the units you entered (cubed).
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For an ellipsoid enter three semi-axes a, b, c; for a cone enter base radius and height.
Frequently asked questions
Why is cone volume one-third of cylinder volume?
A cone with the same base radius and height as a cylinder fills exactly one-third of it — a fact Archimedes proved using the method of exhaustion. Intuitively, think of pouring three full cones of liquid into a cylinder of identical dimensions: it fills exactly.
What is the difference between a pyramid and a cone?
A pyramid has a polygonal base (square, rectangular, triangular, etc.) and triangular faces meeting at an apex. A cone has a circular base with a smooth curved surface to the apex. Both share the ⅓ factor in their volume formula: V = ⅓ × base area × height.
How do I convert volume between units?
Volume scales as the cube of the linear unit. 1 m = 100 cm, so 1 m³ = 1,000,000 cm³. 1 foot = 12 inches, so 1 ft³ = 1,728 in³. Common liquid conversions: 1 liter = 1,000 cm³; 1 gallon (US) ≈ 3,785 cm³. Always cube the linear conversion factor.
What is an ellipsoid and where does it appear in nature?
An ellipsoid is a 3D generalization of an ellipse — a stretched or squashed sphere. The Earth is an oblate spheroid (a flattened ellipsoid, with equatorial radius slightly larger than polar radius). Eggs, watermelons, and cell nuclei are approximately ellipsoidal. The ellipsoid volume formula (4/3)πabc reduces to (4/3)πr³ when a = b = c = r (perfect sphere).
Volume calculator for all common 3D shapes
Understanding volume formulas
Volume measures the amount of 3D space a solid occupies, expressed in cubic units. For prisms (cube, box, cylinder), volume = base area × height — every horizontal slice is identical. For pyramids and cones, which taper to a point, the average cross-section is one-third the base area, giving the ⅓ factor. The sphere formula (4/3)πr³ is derived by integration or by Archimedes' displacement argument.
Practical applications of volume calculation
Volume calculations govern everyday engineering: tank capacity (cylinders and boxes), concrete pour volume (rectangular boxes and cylinders), pill dosing (ellipsoids and cylinders), grain silo storage (cylinders and cones), and fuel tank design. Builders calculate the volume of concrete needed for a cylindrical column or a rectangular footing. Farmers estimate grain bin capacity using the cylinder-plus-cone formula for a hopper bin.
The sphere as the most efficient shape
Among all shapes with a given surface area, the sphere encloses the maximum volume — a result from the calculus of variations called the isoperimetric inequality. This is why soap bubbles are spherical (minimizing surface energy for a given air volume), why planets are roughly spherical (gravity pulls mass toward a center), and why pressure vessels in industry are cylindrical or spherical (to distribute stress uniformly).
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