Surface Area of a Hemisphere Calculator
Enter any one known measurement of a hemisphere and this calculator instantly returns all three surface areas (curved cap, flat base, and total), plus the volume and base circumference. Choose your unit, then switch between solving from the radius, diameter, volume, or curved surface area.
Formula
Worked example
For a hemisphere with radius 5 cm: curved area = 2 * pi * 5^2 = 157.08 cm^2; base area = pi * 5^2 = 78.54 cm^2; total = 235.62 cm^2; volume = (2/3) * pi * 5^3 = 261.80 cm^3.
What is the surface area of a hemisphere?
A hemisphere is exactly half of a sphere, created by cutting a sphere through its center. Unlike a full sphere, a hemisphere has two distinct surfaces: the curved dome (called the cap or lateral surface) and the flat circular face at the cut (the base). The curved surface area is 2 * pi * r^2, the base area is pi * r^2, and the total surface area is their sum, 3 * pi * r^2. The base area happens to equal exactly half the curved area, so the curved dome always makes up two-thirds of the total surface.
How to calculate the surface area of a hemisphere
Start with the radius r. The curved cap area is 2 * pi * r^2. This is exactly half the surface area of the original full sphere (4 * pi * r^2). The flat base is just the area of a circle: pi * r^2. Add them together for the total: 3 * pi * r^2. If you only know the diameter, divide it by 2 to get r. If you know the volume, rearrange V = (2/3) * pi * r^3 to get r = (3V / (2 * pi))^(1/3). If you know the curved surface area, r = sqrt(A_cap / (2 * pi)). This calculator accepts any of those starting points and computes the rest instantly.
Curved area vs. total area: which one do you need?
The choice depends on whether the flat base is also a covered surface. For a hemispherical dome or bowl open at the top, only the curved area matters for material estimation. For a solid half-sphere object (like a hemisphere paperweight), the total surface area including the base is the right number. A hemispherical pressure vessel, for example, requires knowing the total area to calculate mass. Painting the outside of a dome uses only the curved area. Both values are always displayed so you can pick the right one.
Real-world applications
Hemispheres appear in architecture (geodesic domes, stadium roofs, igloo shapes), engineering (tank heads, radomes, pressure domes), manufacturing (lens blanks, bowl tooling, hemisphere molds), and everyday objects (salad bowls, half-sphere candles, snow globes). Knowing the surface area lets you estimate how much material, paint, insulation, or coating is needed. Volume is useful for capacity calculations such as water storage or concrete fill. Base circumference helps with fitting a rim or flange around the circular edge.
Common hemisphere surface area reference (radius in cm)
| Radius (cm) | Curved area (cm^2) | Base area (cm^2) | Total area (cm^2) | Volume (cm^3) |
|---|---|---|---|---|
| 1 | 6.2832 | 3.1416 | 9.4248 | 2.0944 |
| 2 | 25.1327 | 12.5664 | 37.6991 | 16.7552 |
| 3 | 56.5487 | 28.2743 | 84.8230 | 56.5487 |
| 5 | 157.0796 | 78.5398 | 235.6194 | 261.7994 |
| 10 | 628.3185 | 314.1593 | 942.4778 | 2094.3951 |
| 15 | 1413.7167 | 706.8583 | 2120.5750 | 7068.5835 |
| 20 | 2513.2741 | 1256.6371 | 3769.9112 | 16755.1608 |
| 25 | 3926.9908 | 1963.4954 | 5890.4863 | 32724.9235 |
| 50 | 15707.9633 | 7853.9816 | 23561.9449 | 261799.3878 |
Total surface area = 3 * pi * r^2. Curved area = 2 * pi * r^2. Base area = pi * r^2.
Frequently asked questions
What is the formula for the total surface area of a hemisphere?
Total surface area = 3 * pi * r^2, where r is the radius. This combines the curved cap (2 * pi * r^2) and the flat circular base (pi * r^2). For example, a hemisphere with r = 5 cm has a total surface area of 3 * pi * 25 = 235.62 cm^2.
Is the curved surface area the same as the total surface area?
No. The curved (cap) surface area is 2 * pi * r^2, which covers only the dome. The total surface area adds the flat circular base (pi * r^2), giving 3 * pi * r^2. The base always adds one-third to the total, so the curved surface is two-thirds of the total area.
How do I find the radius if I only know the surface area?
It depends on which area you know. If you have the total surface area (A_total), rearrange 3 * pi * r^2 = A_total to get r = sqrt(A_total / (3 * pi)). If you have the curved area, r = sqrt(A_cap / (2 * pi)). Enter either value directly in this calculator by choosing the matching input mode.
What is the difference between a hemisphere and a half-sphere?
They are the same shape. A hemisphere is the formal geometric term for exactly half of a sphere, divided along a great circle. The flat face is the base and the rounded outer surface is the cap. Some contexts use half-sphere, dome, or hemispherical cap informally to mean the same thing.
How does the surface area change if I double the radius?
Surface area scales with the square of the radius. Doubling the radius multiplies every area (curved, base, and total) by 4. Tripling the radius multiplies area by 9. Volume scales with the cube, so doubling the radius increases volume by a factor of 8.
Can I calculate the hemisphere surface area from the volume?
Yes. Rearrange V = (2/3) * pi * r^3 to find r = (3V / (2 * pi))^(1/3), then use r to compute all three areas. Select "Volume" in the "Solve from" dropdown above and enter your volume to have this calculator do it automatically.