A hemisphere is exactly half a sphere — but its surface area is NOT half of 4πr². This distinction trips up more students than almost any other geometry concept.
Why This Confusion Exists
When you cut a sphere in half, you create a new surface — the flat circular base. This base adds πr² to the curved dome area. That's why "half a sphere" doesn't mean "half the surface area."
The Two Hemisphere Formulas
Open hemisphere (curved dome only):
SA = 2πr²
Closed hemisphere (dome + flat base):
SA = 2πr² + πr² = 3πr²
SA = 2πr²
Closed hemisphere (dome + flat base):
SA = 2πr² + πr² = 3πr²
Full Sphere vs Hemisphere Comparison
| Measurement | Full Sphere | Open Hemisphere | Closed Hemisphere |
|---|---|---|---|
| Formula | 4πr² | 2πr² | 3πr² |
| r = 5 cm | 314.16 cm² | 157.08 cm² | 235.62 cm² |
| r = 10 cm | 1,256.6 cm² | 628.3 cm² | 942.5 cm² |
Real-World Decision Guide
| Object | Formula | Why |
|---|---|---|
| Dome roof (paint outside) | 2πr² (open) | No flat base exposed |
| Salad bowl (interior coating) | 2πr² (open) | Bowl interior only |
| Half-ball stress toy | 3πr² (closed) | All surfaces exposed |
| Igloo (outside surface) | 2πr² − door opening | Dome minus entrance |
Common Mistake: Forgetting the flat base is the #1 hemisphere error. Always ask: is the flat circular face exposed? If yes → use 3πr². If the base is attached to another surface → use 2πr².
Memory Aid: Sphere = 4πr². Open hemisphere = 2πr² (exactly half the sphere). Closed hemisphere = 3πr² (half the sphere + one circle).
Calculate Hemisphere Surface Area
Use our Sphere Calculator and halve as needed. See sphere derivation (4πr²) and composite shapes for hemisphere-on-cylinder problems.