Sphere — Volume + Surface Area
Sphere: V = (4/3)πr³, SA = 4πr². The most-symmetric solid; same SA = SA of cylinder of equal r and h = 2r.
Result
Volume
523.5988
(4/3)πr³ = 523.5988. SA = 4πr² = 314.1593.
- Radius (r)5
- Diameter (2r)10.000000
- Great-circle circumference (2πr)31.415927
- Volume ((4/3)πr³)523.598776
- Surface area (4πr²)314.159265
Step-by-step
- V = (4/3)πr³ = (4/3) × π × 5³ = 523.5988.
- SA = 4πr² = 4 × π × 25 = 314.1593.
- Diameter = 2r = 10.0000; great-circle circumference = 2πr = 31.4159.
How to use this calculator
- Enter radius.
- Read volume, surface area, and great-circle dimensions.
About this calculator
A sphere is the locus of points equidistant from a center. Volume V = (4/3)πr³. Surface area SA = 4πr². Archimedes proved (~250 BC) that a sphere has exactly ⅔ the volume and ⅔ the surface area of the smallest cylinder containing it (radius r, height 2r) — he asked for this discovery to be on his tomb. The sphere has the smallest SA of any solid for a given volume, which is why bubbles, planets, and liquid drops in zero-g become spherical. Source: Stewart, "Calculus" §6.2; Apostol, "Calculus" Vol II.
Frequently asked
Integrating the area of circular cross-sections π(r² − x²) from −r to +r gives (4/3)πr³. Archimedes did this geometrically without calculus.
Related calculators
Sphere Volume Calculator
V = (4/3)πr³. Surface area = 4πr².
Sphere Surface Area Calculator
SA = 4πr². Exactly 4× the area of a great circle of the same radius.
Hemisphere — Volume + Surface Area
Hemisphere (half sphere): V = (2/3)πr³, curved SA = 2πr², total SA (with flat base) = 3πr².
Cylinder — Volume + Surface Area
Right circular cylinder: V = πr²h, lateral SA = 2πrh, total SA = 2πr(r + h).
Ellipsoid — Volume + Surface Area
Ellipsoid: V = (4/3)π·a·b·c (exact), SA ≈ Knud Thomsen 1.6075 approximation (≤1.061% error worldwide).
Torus — Volume + Surface Area
Torus (donut): V = 2π²·R·r², SA = 4π²·R·r. R = major (center to tube center), r = minor (tube radius).