Volume Calculator
Calculate the volume and surface area of common 3D shapes.
| Shape | Volume Formula |
|---|---|
| Cube | a³ |
| Rect. Prism | l × w × h |
| Sphere | (4/3)πr³ |
| Cylinder | πr²h |
| Cone | (1/3)πr²h |
| Square Pyramid | (1/3)a²h |
| Ellipsoid | (4/3)πabc |
| Capsule | πr²(4r/3 + h) |
Calculate the volume and total surface area of eight common 3D solids — cubes, rectangular prisms, spheres, cylinders, cones, square pyramids, ellipsoids, and capsules — from their dimensions.
How Volume Calculator Works
Each shape in the dropdown uses its own standard geometric formula: a cube's volume is a³, a rectangular prism is l×w×h, a sphere is (4/3)πr³, a cylinder is πr²h, and a cone is (1/3)πr²h — the ⅓ factor reflecting that a cone holds exactly a third of the cylinder that circumscribes it.
Surface area is calculated with a matching shape-specific formula — for instance, a cylinder's surface area (2πr(r+h)) adds the curved side to the two circular ends, while a cone's accounts for the slanted cone surface plus its circular base.
The ellipsoid's surface area has no exact elementary formula, so the calculator uses the widely accepted Knud Thomsen approximation (exponent p ≈ 1.6075), and the capsule — a cylinder capped with two hemispheres — uses V = πr²(4r/3 + h), where the 4r/3 term is exactly the volume contribution of the two hemispherical ends combined into one full sphere.
See It In Action
Who Uses Volume Calculator and Why
- Estimating how much water or liquid a cylindrical or spherical tank can hold from its dimensions.
- Working out the internal volume of a rectangular prism or capsule-shaped container for packaging design.
- Comparing the surface area and volume of a cone or pyramid for a school geometry project.
- Estimating material volume needed for a 3D-printed or molded object across several supported shapes.
Mistakes to Avoid
- Mixing units within one calculation — entering one dimension in centimeters and another in inches makes both the volume and surface area meaningless.
- Entering a diameter value into a field that expects a radius (or the reverse), which throws off every shape's formula since they're all built around radius.
- Expecting the ellipsoid's surface area to be mathematically exact — it's a widely used approximation, not a precise closed-form calculation.
Tips for Best Results
- Keep every dimension in the same unit before entering them — the calculator is unit-agnostic and simply trusts that consistency.
- For a true capsule shape (a cylinder with two hemispherical caps), remember the extra 4r/3 term in its volume formula already accounts for both rounded ends combined into one sphere's worth of volume.
Fixing Common Problems
My ellipsoid surface area doesn't exactly match a value from another source. — This is expected — there's no exact elementary formula for an ellipsoid's surface area, so the calculator uses the Knud Thomsen approximation, accurate to within roughly 1%, not an exact figure.
Terms Explained
Knud Thomsen approximation: A formula (using exponent p ≈ 1.6075) for estimating an ellipsoid's surface area, since no exact elementary formula exists.
Capsule: A cylinder capped on both ends by hemispheres, whose volume combines a cylinder's formula with one full sphere's worth of volume from the two caps.