Calculate the volume, surface area, and diameter of a sphere from any known value.
Enter any value — radius, diameter, surface area or volume — and all others are calculated instantly.
cm
r
cm
d = 2r
cm²
A = 4πr²
cm³
V = ⁴⁄₃πr³
What is a sphere?
A sphere is a perfectly round three-dimensional geometric solid. Every point on its surface is at the same distance — the radius r — from a fixed central point. Unlike a circle, which is a flat 2D curve, a sphere occupies three-dimensional space.
In a 3D coordinate system centred at the origin, the equation of a sphere of radius r is:
x² + y² + z² = r²
This means that a point (x, y, z) lies on the sphere if and only if the sum of the squares of its coordinates equals r². From this single equation follow the formulas for surface area (A = 4πr²) and volume (V = ⁴⁄₃πr³).
Frequently asked questions
The volume of a sphere is calculated with V = (4/3)πr³, where r is the radius. Volume grows with the cube of the radius, so doubling the radius multiplies volume by eight. For r = 5 cm, V = (4/3) × π × 125 ≈ 523.6 cm³. Archimedes proved that this equals exactly two-thirds the volume of the smallest cylinder that can contain the sphere.
The surface area of a sphere is the total area of its curved outer layer, calculated with A = 4πr². Notably, this is exactly four times the area of the circle with the same radius. For r = 5 cm, A = 4 × π × 25 ≈ 314.16 cm². To find the radius from the surface area: r = √(A / (4π)).
A circle is a two-dimensional (2D) figure — a flat, perfectly round curve where every point is equidistant from the centre. A sphere is a three-dimensional (3D) object — a perfectly round solid where every point on the surface is equidistant from the centre. Think of a circle as a coin and a sphere as a ball. Their equations are analogous: circle area = πr², sphere volume = (4/3)πr³.
From volume V: r = ∛(3V / (4π)). From surface area A: r = √(A / (4π)). This calculator performs these inverse calculations automatically. For example, a spherical tank holding 1,000 litres has r ≈ 0.620 m, giving a diameter of about 1.24 m.
Sphere geometry appears across many fields. In engineering, spherical tanks are sized using surface-area and volume formulas — a sphere encloses the maximum volume for a given surface area. In sports, ball regulations specify precise diameters. In astronomy, planets are modelled as spheres to estimate their mass. In medicine, spherical approximations estimate tumour and organ sizes from scan measurements.