Sphere Volume Calculator
Compute every dimension of a sphere from any one known value: radius, diameter, volume, surface area, or circumference. Switch to hemisphere or spherical cap mode, pick your length unit, and get the answer with a full show-your-work panel.
Formula
Worked example
For a sphere of radius 5 cm: V = (4/3) * pi * 125 = 523.5988 cm^3, A = 4 * pi * 25 = 314.1593 cm^2, C = 2 * pi * 5 = 31.4159 cm. Knowing only the circumference 31.4159 cm: r = 31.4159 / (2*pi) = 5 cm, same answer. For a spherical cap cut 2 cm deep from the same 5 cm sphere: V = (pi * 4 / 3) * (15 - 2) = 54.45 cm^3.
How the sphere volume formula works
A sphere is defined as the set of all points exactly a distance r (the radius) from a fixed centre. Its volume is V = (4/3) * pi * r^3. Because r is cubed, volume scales dramatically with size: double the radius and the volume increases eightfold (2^3 = 8). The formula was proved by Archimedes, who showed that a sphere occupies exactly two-thirds of the volume of the smallest cylinder that can contain it. If you only have the diameter, divide by two before cubing; using the diameter directly would overstate the volume by a factor of eight.
Surface area, circumference, and the relationships between them
The surface area A = 4 * pi * r^2 is exactly four times the area of a circle with the same radius. It also equals the derivative of the volume with respect to r, meaning each thin spherical shell added to the outside has an area of 4 * pi * r^2. The circumference C = 2 * pi * r is the perimeter of any great circle (the largest circle that fits inside the sphere, such as the equator of a globe). These three formulas are tightly linked: knowing any one of radius, volume, surface area, or circumference lets you derive all the others, which is why this calculator offers reverse-solve modes for volume and surface area.
Hemisphere and spherical cap modes
A hemisphere is exactly half a sphere, so its volume is V = (2/3) * pi * r^3 and its total surface area includes both the curved dome (2 * pi * r^2) and the flat circular base (pi * r^2). A spherical cap is the region cut from a sphere by a flat plane; it is defined by the sphere radius r and the cap height h (the perpendicular distance from the cutting plane to the apex). Its volume is V = (pi * h^2 / 3) * (3r - h) and its curved surface area is 2 * pi * r * h. The cap height must be at most 2r (a full hemisphere is the special case h = r; a full sphere is h = 2r).
Using the calculator: unit system and reverse-solve
Choose a length unit (mm through km, or in/ft/yd) and all inputs and outputs use that unit consistently, so volumes are always in cubic units and areas in square units of your chosen length. The "I want to calculate" selector unlocks reverse-solve: enter a known volume to find the radius, or enter a known surface area to work backwards to the radius. The circumference mode is practical when you can only wrap a tape measure around the outside of an object and cannot measure through its centre. Toggle "Show result as a multiple of pi" to see the exact rational coefficient alongside the decimal for use in hand-calculations or exams.
Sphere properties by radius (same unit throughout)
| Radius (r) | Diameter (d) | Circumference (C) | Surface area (A) | Volume (V) |
|---|---|---|---|---|
| 1 | 2 | 6.2832 | 12.5664 | 4.1888 |
| 2 | 4 | 12.5664 | 50.2655 | 33.5103 |
| 3 | 6 | 18.8496 | 113.0973 | 113.0973 |
| 5 | 10 | 31.4159 | 314.1593 | 523.5988 |
| 7 | 14 | 43.9823 | 615.7522 | 1436.7550 |
| 10 | 20 | 62.8318 | 1256.6371 | 4188.7902 |
| 15 | 30 | 94.2478 | 2827.4334 | 14137.1669 |
| 20 | 40 | 125.6637 | 5026.5482 | 33510.3216 |
Computed with V = (4/3)*pi*r^3, A = 4*pi*r^2, C = 2*pi*r.
Frequently asked questions
What is the formula for the volume of a sphere?
The volume is V = (4/3) * pi * r^3, where r is the radius. Cube the radius, multiply by pi (about 3.14159), then multiply by four-thirds. For example, r = 5 cm gives (4/3) * pi * 125 = 523.60 cm^3.
How do I find the volume if I only know the diameter?
Halve the diameter to get the radius, then use V = (4/3) * pi * r^3. A 10 cm diameter gives r = 5 cm and V = 523.60 cm^3. Use the "Volume (I know the diameter)" mode and the calculator halves it for you automatically.
How do I calculate the radius from the volume?
Rearrange V = (4/3) * pi * r^3 to get r = (3V / (4 * pi))^(1/3). Select "Radius (I know the volume)", enter the volume, and the calculator finds the radius, surface area, and circumference in one step.
Can I find the volume from the circumference?
Yes. The circumference of the great circle is C = 2 * pi * r, so r = C / (2 * pi). Once you have the radius, use V = (4/3) * pi * r^3 as usual. This is useful for round objects like balls, where wrapping a tape measure around the outside is easier than measuring through the centre. Select the "circumference" mode to do this automatically.
What is a spherical cap and how is its volume calculated?
A spherical cap is the rounded piece cut from a sphere by a flat plane. It is defined by the sphere radius r and the cap height h. Its volume is V = (pi * h^2 / 3) * (3r - h) and its curved surface area is A = 2 * pi * r * h. A hemisphere is the special case where h = r.
Why is the surface area of a sphere 4 * pi * r^2?
A = 4 * pi * r^2 is exactly four times the area of a circle with the same radius. It is also the derivative of the volume V = (4/3) * pi * r^3 with respect to r, meaning each infinitely thin shell you add to the outside of a sphere has an area of 4 * pi * r^2. This connection through calculus is why surface area and volume are so tightly linked.
Does the unit matter when using this calculator?
The unit matters for the magnitude of the result but not the method. Choose cm, m, in, ft, or any other supported unit: the formulas are the same and the outputs are always in the correct cubic or square units of your chosen length. Just be consistent; do not mix cm and inches in the same calculation.