Geometry desk
Sphere Surface Area & Volume Calculator
Calculate a sphere's surface area and volume from its radius.
Sphere Surface Area & Volume Calculator: Get both the outside surface area and the space enclosed by a complete sphere from its radius. The two results separate how much surface could be covered from how much volume the ideal ball occupies, with square and cubic units shown together. Runs 100% locally in your browser with zero server file uploads.
- Category
- Calculators
- Runs
- In your browser
- Cost
- Free · no sign-up
- Availability
- Ready to use
Surface area
314.16 cm²
- Volume
- 523.6 cm³
Surface area = 4πr²; Volume = 4πr³ ÷ 3- Dimensions are measured in the same unit.
- Values must be finite and greater than zero.
Formula and worked example
Surface area = 4πr²; Volume = 4πr³ ÷ 3
A worked example with sample values:
| Inputs | |
|---|---|
| Radius | 5 cm |
| Results | |
|---|---|
| Surface area | 314.16 cm² |
| Volume | 523.6 cm³ |
Surface and enclosed space use different powers
Let r be radius, S surface area and V volume. S = 4 × π × r², while V = 4 × π × r³ ÷ 3. The surface covers four times the area of a circle of radius r. To understand the volume, slice the sphere into discs: a slice at distance x from its centre has area π(r² − x²). Summing these thin slices from −r to r gives 4πr³ ÷ 3.
At radius 5 cm, S = 4 × π × 25 = 100π = 314.159265… cm², displayed as Surface area = 314.16 cm². Volume is V = 4 × π × 125 ÷ 3 = 523.598775… cm³, displayed as Volume = 523.6 cm³.
OpenStax's solid-geometry lesson (https://openstax.org/books/prealgebra-2e/pages/9-6-solve-geometry-applications-volume-and-surface-area) lists both sphere formulas. The slicing approach is developed in its calculus text.
Scaling and converting a ball
Doubling a ball's radius increases its outside surface fourfold but enclosed space eightfold. Neither output doubles. Conversions follow the same distinction: 1 m² equals 10,000 cm², whereas 1 m³ equals 1,000,000 cm³. Divide the example volume by 1,000 to obtain about 0.523599 litres. NIST's SI guide (https://www.nist.gov/pml/special-publication-811/nist-guide-si-chapter-4-two-classes-si-units-and-si-prefixes) identifies square metres and cubic metres as area and volume units. The unit menu does not convert an already entered radius.
How to use it
- Enter positive dimensions in the same unit.
- Choose the unit used for every dimension.
- Calculate, then review the result and formula.
Privacy & limitations
Every value and calculation stays in your browser. No dimensions are uploaded or sent to an external service.
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Frequently asked questions
Is the sphere radius half the measured diameter?
Yes. A ball measuring 10 cm across has radius 5 cm. Using its diameter as radius multiplies the surface area by four and volume by eight, so confirm which measurement you have before entering it.
How do I find a sphere's radius from its volume?
Rearrange V = 4πr³ ÷ 3 to r = ∛(3V ÷ (4π)). For volume 500π ÷ 3 cm³, the expression inside the cube root is 125 cm³, giving radius 5 cm. Surface area instead gives r = √(S ÷ (4π)).
Does the sphere volume mean a hollow ball's material volume?
No. It is the full volume within the supplied radius. For a hollow shell, calculate the outer and inner sphere volumes separately and subtract. The inner radius must be smaller, and the two spheres must describe the intended shell.
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