Library / Geometry, coordinates, surveying and trigonometry / GEO-19
Geometry, coordinates, surveying and trigonometry
Surface area of a sphere
Area of the whole curved surface of a sphere from its radius.
- Ref
- GEO-19
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- S [cm²]
The curved surface of a sphere of radius r has an area of 4πr², exactly four times the area of the circle through its centre. Archimedes showed it equals the curved surface of the cylinder that just encloses the sphere. For a hemisphere, say whether the flat base is included: add πr² if it is.
Area of the whole curved surface of a sphere from its radius. It is four times the area of the great circle.
Calculator
- Volume, 4πr³/3113.1 cm³
- Area of a hemisphere with its flat base, 3πr²84.82 cm²
- Surface per unit volume, 3/r1 1/cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A sphere of radius 3 cm has 113.1 cm² of surface and 113.1 cm³ of volume.
- Doubling the radius gives four times the surface and eight times the volume, so small spheres lose heat faster for their size.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Radius | 0.5 to 30 cm | 3.142 | 11,310 | 11,310 | +10% | +21 % |
Sweep of Radius
| Radius (cm) | Surface area (cm²) |
|---|---|
| 0.5 | 3.142 |
| 4.188 | 220.4 |
| 7.875 | 779.3 |
| 11.56 | 1,680 |
| 15.25 | 2,922 |
| 18.94 | 4,507 |
| 22.63 | 6,433 |
| 26.31 | 8,700 |
| 30 | 11,310 |
Sweep one variable
Variables and how each one contributes
Surface area
What the equation gives you. Everything else feeds it.
Radius
The radius of the sphere, in cm. Surface area grows with the square of the radius, so a 10 % increase gives a 21 % increase. Raising it by 10 % (to 3.3) moves the result by +21 %, from 113.1 to 136.8 cm².
Worked example with the values above
- Square the radius: r² = 3² = 9.
- Multiply by 4π = 12.566.
- S = 113.1 cm².
Assumptions and limits
- Solids with flat or smooth faces and a uniform cross-section where stated.
- All lengths in the same unit, with areas and volumes in the matching square and cubic units.
Common mistakes
- Using the volume formula 4πr³/3 by mistake.
- Forgetting that a hemisphere also needs its flat base area if the base is included.
- Using the diameter in place of the radius, which gives four times too much.
Statistics and uncertainty
Surface area scales with the square of the radius, so a 1 % error in r gives about a 2 % error in S.
Where it is used
- Material, paint and coating quantities for bars, tanks, hoppers and domes.
- Estimating volumes, masses and capacities from drawings.
Maths you need
- Volume: the space inside a solid, in cubic units
- Cross-section: the shape you see cutting the solid
- Surface area: the total area of all outer faces