Surface Area of a Sphere Calculator
Enter any one measurement of a sphere and this calculator instantly works out the surface area, volume, diameter, radius, circumference, and surface-to-volume ratio. Switch between metric and imperial units and see each step of the working shown below the result.
Formula
Worked example
For a sphere with radius 5 cm: A = 4 × π × 5² = 4 × π × 25 = 314.159 cm². Volume = (4/3) × π × 125 = 523.599 cm³. Circumference = 2 × π × 5 = 31.416 cm.
What is the surface area of a sphere?
The surface area of a sphere is the total area of its outer curved surface. Unlike a cube or cylinder, a sphere has no flat faces or edges: every point on its surface is equidistant from the centre, which gives it the simplest possible geometry. The formula A = 4πr² was first proved by Archimedes around 250 BCE. He showed that the surface area of a sphere equals exactly four times the area of its great circle (the largest circle that fits inside it), and that it also equals the lateral surface area of the smallest cylinder that can contain it. That elegant relationship is why the constant 4π appears in the formula.
How to use this calculator
Choose the measurement you already know from the "Known measurement" dropdown: radius, diameter, circumference, volume, or surface area. Enter the value, pick your unit system (metric centimetres or imperial inches), and all five other sphere measurements are computed instantly. The "Show your work" panel below the result traces every step of the algebra, including how the radius is derived from your chosen input before the surface area is worked out.
Deriving surface area from any sphere measurement
Because all sphere measurements are functions of a single parameter, the radius, you only ever need one of them to find the rest. From diameter: r = d/2, then A = 4πr². From circumference: r = C/(2π), then A = 4πr². From volume: r = ∛(3V/4π), then A = 4πr². From surface area directly: A is already known, but r = √(A/4π) is worked out as a check. Equivalently, A = πd² (from diameter), A = C²/π (from circumference), and A = ∛(36πV²) (from volume) are compact shortcut formulas when you want to skip deriving the radius first.
Surface-to-volume ratio and why it matters
The surface-to-volume ratio (A/V = 3/r for a sphere) describes how much surface area a shape has relative to its size. A smaller sphere has a higher ratio, meaning proportionally more surface relative to its interior. This number matters in biology (cells must exchange nutrients across their surface), engineering (heat exchangers, catalysts), and everyday physics (why small raindrops fall more slowly, why smaller ice cubes melt faster). Among all closed three-dimensional shapes, a sphere has the lowest possible surface-to-volume ratio for a given volume, which is why pressurised vessels and soap bubbles are spherical.
Surface area and volume for common sphere sizes
| Radius (cm) | Diameter (cm) | Surface area (cm²) | Volume (cm³) |
|---|---|---|---|
| 1 | 2 | 12.566 | 4.189 |
| 2 | 4 | 50.265 | 33.510 |
| 3 | 6 | 113.097 | 113.097 |
| 5 | 10 | 314.159 | 523.599 |
| 7 | 14 | 615.752 | 1,436.755 |
| 10 | 20 | 1,256.637 | 4,188.790 |
| 15 | 30 | 2,827.433 | 14,137.167 |
| 20 | 40 | 5,026.548 | 33,510.322 |
| 50 | 100 | 31,415.927 | 523,598.776 |
| 100 | 200 | 125,663.706 | 4,188,790.205 |
Values computed from A = 4πr² and V = (4/3)πr³.
Frequently asked questions
What is the formula for the surface area of a sphere?
The surface area of a sphere is A = 4πr², where r is the radius. Equivalently, A = πd² when you know the diameter. These formulas were first proved by Archimedes, who showed that the surface equals four great-circle areas.
How do I find surface area from volume?
First derive the radius from the volume using r = ∛(3V/4π), then substitute into A = 4πr². Alternatively, use the shortcut A = ∛(36πV²) to go directly from volume to surface area without finding the radius as an intermediate step. This calculator does all of that for you when you select "Volume" in the known-measurement dropdown.
What is the difference between surface area and volume of a sphere?
Surface area (A = 4πr²) measures the outer two-dimensional skin of the sphere, expressed in square units. Volume (V = 4/3 πr³) measures the three-dimensional space enclosed inside it, expressed in cubic units. Volume grows with the cube of the radius, so large spheres have proportionally much more interior than surface.
What units does this calculator use?
You can work in either metric (centimetres) or imperial (inches). Switch the unit system before entering your value. The surface area is then returned in cm² or in², the volume in cm³ or in³, and the radius, diameter, and circumference in the chosen linear unit. The surface-to-volume ratio is expressed per cm or per inch.
How is the circumference of a sphere defined?
A sphere has infinitely many circles on its surface. The "circumference" referred to here is the great circle circumference: the distance around the largest possible cross-section, which passes through the centre. Its formula is C = 2πr, the same as for any circle with that radius.
Can I calculate sphere measurements from the surface area?
Yes. Select "Surface area" in the known-measurement dropdown, enter the area, and the calculator derives the radius as r = √(A/4π), then computes volume, diameter, and circumference from that radius.