Surface Area of a Hemisphere Calculator
Get the surface area of a half-sphere from its radius or diameter, in any unit. You get all three numbers people mean by it: the curved dome (2πr²), the flat circular base (πr²), and the closed total (3πr²).
Example: with Measurement Radius · Value 5 → Total surface area (3πr²): 235.6 square units.
- Curved (dome) area (2πr²)157.1 square units
- Flat base circle (πr²)78.54 square units
Computed by the calculator below using its default values. Change any input to see your own numbers.
Curved half-sphere = 2πr² (half of a sphere's 4πr²); add the base circle πr² for a closed solid: 3πr² total.
Why a hemisphere is 3πr², not 2πr²
Cut a sphere in half and the curved skin splits evenly: each half carries 2πr², half of the sphere's total 4πr². But cutting also exposes a brand-new flat face — a circle of area πr² — that was not part of the original surface. A closed hemisphere (dome plus floor) therefore totals 2πr² + πr² = 3πr². Which number you need depends on the job: painting just the dome uses 2πr², wrapping the whole solid uses 3πr².
A neat consequence of Archimedes' sphere result: the curved part alone (2πr²) exactly equals the lateral area of the cylinder that would wrap around it. And the dome holds surprising area — a 5-unit-radius dome carries 157 square units of curved surface over a floor of only 78.5, which is why dome roofs cost more per square foot of footprint than flat ones.
How it’s calculated
With radius r (r = diameter/2 when entering a diameter): curved surface = 2πr²; base circle = πr²; total closed surface = 3πr². π ≈ 3.14159265. Results are in the square of whatever unit you enter.
A true half-sphere cut through the center — shallow domes and stretched shells need the spherical-cap formula 2πrh instead.
Hemisphere surface areas
| Radius | Curved 2πr² | Base πr² | Total 3πr² |
|---|---|---|---|
| 1 | 6.28 | 3.14 | 9.42 |
| 2 | 25.13 | 12.57 | 37.70 |
| 5 | 157.08 | 78.54 | 235.62 |
| 10 | 628.32 | 314.16 | 942.48 |
Computed with 2πr², πr², and 3πr²; rounded to 2 decimals.
Common mistakes
- Assuming a hemisphere's surface is half a sphere's — the curved part is, but the exposed base circle adds another πr², making the closed total 3πr².
- Entering a diameter as the radius, which inflates every area by 4× since r is squared.
- Using 3πr² when you only need to coat the dome; paint estimates want the curved 2πr² alone.
- Confusing surface area (square units) with volume — capacity is (2/3)πr³, a different formula on a different page.
Frequently asked questions
What is the surface area of a hemisphere?
Total = 3πr²: the curved dome contributes 2πr² (half a sphere's 4πr²) and the flat circular base adds πr². With r = 5 that is 235.62 square units.
Is a hemisphere's surface area half of a sphere's?
No — that is the classic trap. Halving the sphere halves the curved skin to 2πr², but the cut exposes a new πr² disk. Closed hemisphere total: 3πr², which is 75% of the full sphere's area.
Which formula do I use to paint a dome?
Just the curved part, 2πr². A dome with a 10-ft radius has 628 sq ft of paintable curve — the 314 sq ft floor circle only matters if you are coating it too.
How does the area change with size?
With the square of the radius: doubling r quadruples every area. That is why the r = 10 row (942.5) is 4× the r = 5 row (235.6).