Sphere Calculator
Volume and surface area, and why one is the derivative of the other.
Volume and surface area, and why one is the derivative of the other. Volume is 4πr³⁄3 and surface area is 4πr².
Volume (units³)
113.0973
Surface area 113.0973 units²
Volume is 4πr³⁄3 and surface area is 4πr². The surface is exactly the derivative of the volume with respect to radius, which is not a coincidence: growing a sphere by a hair adds a shell of exactly its own surface area.
How the Sphere Calculator works
Volume is 4πr³⁄3 and surface area is 4πr². The surface is exactly the derivative of the volume with respect to the radius, which is not a coincidence: growing a sphere by a hair adds a shell of precisely its own surface area.
Also known as: volume of a sphere · surface area of a sphere · ball volume calculator · sphere volume formula
Where to go next
The Sphere question rarely arrives on its own. These are the ones that usually come with it:
- Volume Calculator — Six shapes, with the litre conversion built in.
- Surface Area Calculator — Six shapes, and the ratio that explains why small things cool fast.
- Cylinder Calculator — Volume, surface area, and the label that peels off flat.
- Percentage Calculator — Every common percentage question in one place.
Frequently asked questions
How do I find the volume of a sphere?
Four thirds of π times the radius cubed. A radius of 3 gives 36π, or 113.10 cubic units.
What is the surface area of a sphere?
4πr². At a radius of 3 this happens to equal the volume numerically (113.10) which is a quirk of that particular radius, not a general rule.
Why is the volume formula 4/3?
It falls out of integrating the area of circular cross-sections along the diameter. Archimedes found it geometrically: a sphere is exactly two thirds of the cylinder that encloses it.
How much does doubling the radius change things?
Surface area quadruples and volume goes up eightfold. Volume scales with the cube of size, which is why large spheres hold far more than intuition suggests.
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