A sphere is the set of all points at the same distance from a center: a ball bearing, a soap bubble, a globe, a round storage tank. That one distance, the radius, fixes everything else about it. This sphere calculator starts from whatever you can actually measure — radius, diameter, circumference, surface area or volume — and returns all the other measurements, plus the capacity in liters and US gallons.
How to use the sphere calculator
- Under I know the, choose Radius, Diameter, Circumference, Surface area or Volume.
- Type the number in Value. Surface area is read in square units and volume in cubic units of the length unit you pick.
- Choose the Units: mm, cm, m, km, in, ft, yd or mi.
- The headline shows the volume, or the radius if you started from a volume. Below it you get the radius, diameter, circumference, surface area, volume, capacity in liters and US gallons, and the great-circle area. The steps show each formula with your numbers.
Sphere formulas
Working backwards to the radius:
| You know | Radius |
|---|---|
| Diameter d | r = d ÷ 2 |
| Circumference C | r = C ÷ 2π |
| Surface area S | r = √(S ÷ 4π) |
| Volume V | r = ∛(3V ÷ 4π) |
A great circle is any slice through the center, like the equator on a globe. Its area, πr², is exactly one quarter of the sphere’s surface: the peel of an orange would cover four circles the size of its widest cross-section.
Worked example: the air inside a basketball
A regulation NBA basketball has a circumference of 29.5 inches. How much air does it hold, and how much material covers it?
- Choose Circumference, enter 29.5 and set the units to inches.
- Radius: 29.5 ÷ 2π ≈ 4.70 in, so the diameter is about 9.39 in.
- Volume: ⁴⁄₃ × π × 4.695³ ≈ 433.5 in³, which is about 7.10 liters or 1.88 US gallons.
- Surface area: 4 × π × 4.695² ≈ 277.0 in², just under 2 ft² (divide by 144).
These are outside dimensions. The air space is slightly smaller because the wall has thickness.
Why volume grows so fast: the r³ effect
Volume depends on the cube of the radius and surface area on its square. Every time the radius doubles, the surface is multiplied by 4 and the volume by 8:
| Radius | Surface area | Volume |
|---|---|---|
| 1 cm | 12.57 cm² | 4.19 cm³ |
| 2 cm | 50.27 cm² | 33.51 cm³ |
| 4 cm | 201.06 cm² | 268.08 cm³ |
| 8 cm | 804.25 cm² | 2,144.66 cm³ |
The ratio of volume to surface, V ÷ S, works out to r ÷ 3, so it grows with size. That is why big spherical tanks are economical: a tank 20 ft across holds about 4,188.8 ft³, roughly 31,334 US gallons, inside a shell of only 1,256.6 ft². A sphere also encloses the most volume for a given surface area of any shape, which is why soap bubbles and small droplets pull themselves round.
The cube works against you when measuring. A 1% error in the diameter becomes roughly a 3% error in the volume, so measure carefully, ideally the circumference with a flexible tape.
Archimedes and the 2 : 3 ratio
Archimedes of Syracuse, in the third century BC, compared a sphere with the cylinder that just fits around it: same radius, height 2r. The sphere fills exactly two thirds of the cylinder’s volume (⁴⁄₃πr³ against 2πr³), and its surface is two thirds of the cylinder’s total area including both ends (4πr² against 6πr²). The sphere’s surface also equals the cylinder’s curved side on its own. He valued this result so highly that he asked for a sphere inside a cylinder to be carved on his tomb, and Cicero later reported finding the overgrown monument in Syracuse.
You can check the ratio yourself: the cylinder calculator with r = 5 cm and h = 10 cm gives 785.40 cm³, and this calculator gives 523.60 cm³ for r = 5 cm.
Common mistakes
- Using the diameter as the radius. Because the radius is cubed, the volume comes out 8 times too large.
- Typing liters or gallons into Value. Volume must be in cubic units of the chosen length. Convert first: 1 L = 1,000 cm³ and 1 gal = 231 in³.
- Converting cubic units like lengths. 1 ft³ is 1,728 in³ (12 × 12 × 12), not 12 in³.
For half a sphere, such as a dome or bowl, use the hemisphere calculator. A pill shape with rounded ends is handled by the capsule calculator.
Frequently asked questions
What is the formula for the volume of a sphere?
V = 4/3 × π × r³, where r is the radius. A ball with a 5 cm radius holds 4/3 × π × 125 ≈ 523.6 cm³, a little over half a liter.
How do I find the radius of a sphere from its volume?
Reverse the volume formula: r = ∛(3V ÷ 4π). Choose Volume under 'I know the' and the calculator takes the cube root for you. A sphere that holds exactly 1 liter (1,000 cm³) has a radius of about 6.20 cm.
Why does doubling the radius make the volume eight times bigger?
Volume depends on the radius cubed, and 2³ = 8. Surface area depends on the radius squared, so it grows only four times. That is why a large tank holds far more liquid per square foot of steel than a small one.
How do I measure the radius of a real ball?
Wrap a tape measure around the widest part to get the circumference, then choose Circumference in the calculator. That is usually more accurate than measuring the diameter with a ruler, because it is hard to line a ruler up with the center of a curved object.
How are the liters and gallons worked out?
The volume is converted using 1 liter = 1,000 cm³ and 1 US gallon = 231 in³ exactly (about 3.785 liters). The result is the capacity of the whole sphere, so use inside measurements if you want to know what a hollow tank or ball can hold.