Sphere Surface Area Calculator: from radius, diameter, circumference or volume
Find the surface area of a sphere or hemisphere with SA = 4πr² from its radius, diameter, circumference or volume, or work back from the area to the radius. Shows the answer in terms of π, the volume, great-circle size and the steps.
How it works
Sphere: SA = 4πr² = πd² · Hemisphere: curved = 2πr², total with base = 3πr² · r = d ÷ 2 = C ÷ 2π = ∛(3V ÷ 4π) = √(SA ÷ 4π) · Volume = ⁴⁄₃πr³ (hemisphere ⅔πr³)
A sphere’s surface area is 4πr²: exactly four times the area of a great circle, the flat disc πr² you get by slicing through the centre. Archimedes proved it by showing the sphere has the same area as the curved side of the cylinder that just wraps it (2πr × 2r). Written with the diameter, SA = πd². You do not need the radius to start. From a diameter, halve it; from a circumference (a tape measure around the widest part), divide by 2π; from a volume, r = ∛(3V ÷ 4π). Choose “Surface area” to go the other way and find the radius from a known area. A hemisphere has half the curved area, 2πr², and a closed hemisphere (like a dome with a floor) adds its circular base πr² for a total of 3πr². Area grows with the square of the radius and volume with the cube, so doubling a ball’s size gives 4 times the surface and 8 times the volume; the scaling table shows this for your sphere. All lengths must be in the same unit; areas come out in that unit squared. The answer is also given as a multiple of π, which is exact when your inputs are exact.
Worked example
For a radius of 5 (the default): surface area = 4 × π × 5² = 4 × π × 25 = 100π ≈ 314.1593 square units. The volume is ⁴⁄₃ × π × 5³ ≈ 523.5988 cubic units, the diameter is 10, the great-circle circumference is 2 × π × 5 ≈ 31.4159 and the great-circle area is 25π ≈ 78.5398, a quarter of the surface. As a hemisphere, the same radius gives 2π × 25 ≈ 157.0796 of curved area plus a 78.5398 base, 75π ≈ 235.6194 in total.
Sources: CalculatorSoup: Sphere Calculator (formulas for r, V, A, C) · Calculator.net: Surface Area Calculator (ball example, r = 0.325 in) · Math is Fun: Sphere (r = 5 example) · Sphere (Wikipedia)
Frequently asked questions
What is the formula for the surface area of a sphere?
SA = 4πr², where r is the radius. With the diameter it is SA = πd². A ball of radius 5 cm has 4 × π × 25 = 100π ≈ 314.16 cm² of surface.
How do I find the surface area from the circumference?
Divide the circumference by 2π to get the radius, then use 4πr². Shortcut: SA = C² ÷ π. A ball measuring 70 cm around has r = 70 ÷ 2π ≈ 11.14 cm and SA = 4,900 ÷ π ≈ 1,559.7 cm².
How do I find the radius from the surface area?
Rearrange SA = 4πr² to r = √(SA ÷ 4π). For SA = 314.16, r = √(314.16 ÷ 12.566) = √25 = 5. Choose “Surface area” in the first box to do this automatically.
What is the surface area of a hemisphere?
The curved part is half a sphere, 2πr². A closed hemisphere also has a flat circular base of πr², so its total surface area is 3πr². For r = 5 that is 75π ≈ 235.62 square units.
Why is a sphere’s surface exactly four great circles?
Archimedes showed that a sphere’s area equals the curved side of the cylinder that just encloses it: circumference 2πr times height 2r = 4πr². That is four times the great-circle disc πr².
Why does surface area grow slower than volume?
Area scales with r² and volume with r³. Double the radius and the surface quadruples while the volume grows eightfold, which is why large spheres hold a lot for their surface and small droplets lose heat or evaporate quickly.
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This calculator is for educational purposes. Double-check important results before acting on them.