Calculate the radius of a sphere from diameter, circumference, surface area, volume, or spherical cap area. Shows all sphere properties with visual comparisons and a reference table.
The radius of a sphere is the distance from the exact center to any point on the surface. It is the single measurement that fully defines a sphere — unlike cylinders or cones which need two dimensions. From the radius, you can derive every other property: diameter, circumference, surface area, volume, and cross-sectional area.
This calculator lets you find the sphere radius from five different inputs: diameter, great circle circumference, surface area, volume, or the area and height of a spherical cap. Choose the method that matches the data you have, enter the value, and get instant results for all sphere properties.
Spheres are one of the most efficient shapes in nature. For a given volume, a sphere has the smallest possible surface area — which is why bubbles, planets, and water droplets are all roughly spherical. This surface-to-volume efficiency is captured by the ratio V/SA = r/3, which increases linearly with the radius.
The key formulas are: circumference C = 2πr, surface area SA = 4πr², and volume V = (4/3)πr³. The volume formula involves r cubed, which means even a small change in radius significantly affects the volume. For example, doubling the radius increases the volume by a factor of 8, while the surface area only increases by a factor of 4.
Use the preset buttons to explore familiar spherical objects like ping pong balls, tennis balls, basketballs, and even the Earth and Moon. The reference table shows common sphere sizes for quick comparisons, and the visual bars illustrate how the radius relates to the diameter.
A sphere is completely determined by its radius, so converting from diameter, surface area, volume, or cap measurements into radius is often the fastest way to understand the whole object. That makes this tool practical for ball sizing, tank and dome estimates, astronomy examples, and classroom problems where one measured property needs to unlock every other sphere dimension.
r = d/2 | r = C/(2π) | r = √(SA/(4π)) | r = ∛(3V/(4π)) | R = A_cap/(2πh), where d = diameter, C = circumference, SA = surface area, V = volume, A_cap = spherical cap area, h = cap height.
Result: Radius = 5 cm
Using the volume mode with V = 523.6 cm^3, the calculator computes r = cbrt(3V / (4pi)) = cbrt(1570.8 / 12.5664) = cbrt(124.999) = 5 cm. That sphere has diameter 10 cm, great-circle circumference about 31.416 cm, and surface area about 314.1598 cm^2.
A sphere of radius r has **surface area** A = 4πr² and **volume** V = (4/3)πr³. These two formulas completely characterize the sphere, and this calculator inverts them to recover r from whichever quantity you know. From surface area: r = √(A / (4π)). From volume: r = (3V / (4π))^(1/3). From diameter: r = d/2. From circumference of a great circle: r = C / (2π).
The surface-area and volume formulas have an elegant relationship: dV/dr = 4πr² = A. In other words, the rate of increase of the sphere's volume with respect to its radius equals its surface area. This is the 3-D analogue of the fact that d/dr(πr²) = 2πr — differentiating the circle's area gives its circumference.
The **isoperimetric inequality** in 3-D states that among all closed surfaces enclosing a fixed volume, the sphere has the smallest surface area. Equivalently, for a fixed surface area, the sphere encloses the largest volume. This is why soap bubbles (minimizing surface energy at constant enclosed air volume) are spherical, and why biological cells that maximize volume-to-surface-area ratio tend toward spherical shapes.
In **astronomy**, planetary radii are inferred from measured angular diameters and known distances, and from the formula linking surface area to luminosity (stellar radius from the Stefan–Boltzmann law). In **materials science**, the radius of spherical nanoparticles determines surface-to-volume ratio and hence catalytic activity. In **fluid mechanics**, the drag on a sphere is related to its radius by Stokes' law: F = 6πηrv. In **medicine**, tumor volume is approximated as (4/3)πr³ from MRI measurements of the radius. In **sports**, ball dimensions are tightly regulated — a standard basketball has r ≈ 11.9 cm, a tennis ball r ≈ 3.3 cm, a golf ball r ≈ 2.14 cm.
The radius is the distance from the center of the sphere to any point on its surface. It is the only measurement needed to fully define the size of a sphere.
Use r = ∛(3V / (4π)). Multiply the volume by 3, divide by 4π, then take the cube root. For example, V = 523.6 cm³ gives r ≈ 5 cm.
Use r = √(SA / (4π)). Divide the surface area by 4π, then take the square root. SA = 314.16 cm² gives r ≈ 5 cm.
A great circle is the largest circle that can be drawn on a sphere — it passes through the center and divides the sphere into two equal hemispheres. The equator is a great circle of the Earth.
A spherical cap is the portion of a sphere that lies above (or below) a cutting plane. Its area depends on the sphere's radius and the cap height. The formula is A = 2πRh.
Bubbles minimize surface area for a given volume (surface tension). Planets and stars are shaped by gravity pulling equally in all directions toward the center, naturally forming a sphere.