A hemisphere is exactly half a sphere, cut through its center by a flat plane. You meet it as a dome over a building, a mixing bowl, a planetarium ceiling or the rounded end of a pressure tank. Two different areas matter for it, the curved dome and the flat circular base, so this hemisphere calculator reports both separately along with the volume and the capacity in liters and US gallons.
How to use the hemisphere calculator
- Under I know the, choose Radius, Diameter, Volume, Total area or Curved area.
- Enter the number in Value. Total area includes the flat base; curved area is the dome only.
- Pick the Units. Areas come back in square units and volume in cubic units of that length.
- The headline shows the volume, or the radius if you entered a volume. The list adds radius, diameter, capacity in liters and US gallons, curved surface area, base (circle) area, total surface area and the base circumference.
Hemisphere formulas
| You know | Radius |
|---|---|
| Diameter d | r = d ÷ 2 |
| Volume V | r = ∛(3V ÷ 2π) |
| Total area S | r = √(S ÷ 3π) |
| Curved area | r = √(curved area ÷ 2π) |
Worked example: a mixing bowl and a dome roof
How much does a 10-inch bowl hold? A hemispherical bowl measures 10 in across the inside rim.
- Choose Diameter, enter 10 and set the units to inches. The radius is 5 in.
- Volume: ⅔ × π × 5³ ≈ 261.8 in³, about 4.29 liters or 1.13 US gallons, filled to the brim.
How much paint does a dome need? A hemispherical dome roof is 30 ft in diameter.
- Choose Diameter, enter 30 and set the units to feet.
- Curved surface area: 2 × π × 15² ≈ 1,413.7 ft². At a coverage of 350 ft² per gallon, that is a little over 4 gallons per coat.
- The total surface area, 2,120.6 ft², includes the circular floor plan, which you are not painting. Using it would overbuy paint by 50%.
Working the other way, choose Volume to design a container. A bowl that must hold 4 liters (4,000 cm³) needs an inside radius of about 12.41 cm, so it is about 24.8 cm across.
Curved area or total area?
The calculator gives both because real jobs need different ones:
- Curved area only: painting, roofing or insulating a dome; glazing the inside of a bowl; the fabric for a round tent canopy.
- Total area: a closed solid half-ball, such as a paperweight, a cast concrete bollard cap or a foam model, where the flat face needs finishing too.
The two are always in the ratio 2 : 3, and the curved dome is always exactly twice the area of the flat base. That makes quick estimates easy: a dome over a round room with about 707 ft² of floor, like the 30-foot example, has about 1,414 ft² of roof surface.
Why the volume is two thirds of πr³
There is an elegant way to see the formula without calculus, known as Cavalieri’s principle: two solids with equal cross-sections at every height have equal volumes. Stand the hemisphere on its flat base. At height z, its slice is a circle of area π(r² − z²).
Now take a cylinder with radius r and height r, and carve out a cone whose tip sits at the center of the bottom and whose base is the cylinder’s top. At height z the cone’s slice has radius z, so what is left of the cylinder is a ring of area πr² − πz², the same as the hemisphere’s slice. Their volumes must match:
Check it with the 5-inch bowl: a cylinder with r = h = 5 in holds 392.70 in³, the matching cone 130.90 in³, and the difference is 261.80 in³.
Common mistakes
- Treating a shallow dome as a hemisphere. If the height is less than half the base width, the shape is a spherical cap and these formulas overestimate it.
- Ignoring wall thickness. Capacity comes from the inside radius. For the material in a shell, subtract the inner hemisphere’s volume from the outer one.
- Entering liters as the volume. Convert to cubic units first: 1 L = 1,000 cm³, 1 US gallon = 231 in³.
For a full ball use the sphere calculator; for a tank with rounded ends, try the capsule calculator.
Frequently asked questions
What is the volume of a hemisphere?
V = 2/3 × π × r³, exactly half the volume of a sphere with the same radius. A bowl with a 5 in radius holds 2/3 × π × 125 ≈ 261.8 in³, or about 1.13 US gallons.
What is the difference between curved area and total area?
Curved area is the dome alone, 2πr². Total area adds the flat circular base, πr², for 3πr² in all. Use the curved area for painting or covering a dome and the total area for a closed solid half-ball.
Is the curved surface of a hemisphere really twice the base?
Yes. The curved surface is 2πr² and the flat base is πr², so the dome always has exactly twice the area of the circle it covers. This follows from Archimedes' result that a sphere's surface is 4πr².
Can I use this for a shallow dome or a saucer-shaped bowl?
Only if the height equals the radius, that is, half the width across the base. A shallower dome is a spherical cap, which has less volume and area than a hemisphere of the same base width, so these formulas would overestimate it.
How do I find the radius from a volume in liters?
Convert the liters to cubic units first (1 L = 1,000 cm³), choose Volume under 'I know the', and enter the result with cm as the unit. The calculator solves r = ∛(3V ÷ 2π); a 4-liter bowl needs an inside radius of about 12.4 cm.