Volume of a Hemisphere Calculator
Enter the radius or diameter of a hemisphere and instantly get its volume, curved surface area, base area and total surface area. Flip between metric (mm, cm, m, km) and imperial (in, ft, yd, mi) units, or solve in reverse by entering any one measurement to find the radius. The "Show your work" panel walks through each formula with your exact numbers.
Formula
Worked example
For a hemisphere with radius 5 cm: V = (2/3) x pi x 5^3 = (2/3) x pi x 125 = 261.7994 cm^3. Curved surface area A = 2 x pi x 25 = 157.0796 cm^2. Base area B = pi x 25 = 78.5398 cm^2. Total surface area K = 3 x pi x 25 = 235.6194 cm^2.
What is a hemisphere?
A hemisphere is exactly half of a sphere, created by slicing a sphere through its centre with a flat plane. It has two surfaces: a curved dome (half the full sphere surface) and a flat circular base. Think of a bowl, a dome building, an igloo, or half a ball: all approximate hemispheres. The defining measurement is the radius r, the distance from the centre of the flat face to any point on the curved surface. Every other property follows from r alone.
How to use this calculator
Pick a unit system (metric or imperial) from the dropdown, then choose which measurement you already know from the "Solve from" list:
- Radius or Diameter - the most common starting point. Enter the radius or diameter of the base circle.
- Volume - enter the known volume to find the radius and all areas.
- Curved surface area - enter the dome area to find everything else.
- Base area - enter the area of the flat circle.
- Total surface area - enter the combined area of dome plus base.
The results panel shows volume, radius, diameter, curved surface area, base area and total surface area. The "Show your work" panel lists every arithmetic step with your actual numbers substituted in.
Hemisphere formulas explained
All hemisphere properties derive from one number: the radius r.
- Volume: V = (2/3) pi r^3. Imagine filling a hemisphere-shaped mould: the volume is two-thirds of the sphere formula (4/3)pi r^3.
- Curved surface area: A = 2 pi r^2. The dome is exactly half the full sphere surface (4 pi r^2).
- Base area: B = pi r^2. This is just the area of the flat circular base, a standard circle formula.
- Total surface area: K = A + B = 2 pi r^2 + pi r^2 = 3 pi r^2. Add the dome and the base circle together.
- Diameter: d = 2r. The width across the flat face.
The ratio of volume to total surface area always equals r/3, regardless of the size of the hemisphere. This makes it straightforward to check whether your numbers are consistent: multiply K by r/3 and you should get V.
Reverse calculations: finding the radius from any measurement
If you know one property and need the radius, rearrange the formulas above:
- From volume: r = (3V / 2pi)^(1/3) - take the cube root of three times the volume divided by two pi.
- From curved surface area: r = sqrt(A / 2pi) - divide the dome area by 2 pi and take the square root.
- From base area: r = sqrt(B / pi) - divide by pi and take the square root.
- From total surface area: r = sqrt(K / 3pi) - divide by 3 pi and take the square root.
- From diameter: r = d / 2.
All five reverse modes are built into this calculator: select the measurement you have from the "Solve from" menu, type the value, and the radius and every other property appear instantly.
Real-world applications
Hemisphere calculations appear across many fields. In architecture and civil engineering, dome structures (from the Pantheon in Rome to modern sports arenas) are modelled as hemispheres to estimate material quantities and structural loads. In manufacturing, bowl-shaped moulds and tanks are designed using volume and surface area. In food science and catering, hemispherical portions (dome cakes, scoops of ice cream, round bread rolls) are sized using the volume formula. Scientists use the hemisphere model for geophysical calculations such as the volume of an ocean basin or a crater, and in optics for the coverage area of a hemispherical lens or mirror. Knowing all four properties from one entry is the practical advantage this calculator offers.
Hemisphere properties for common radii (metric)
| Radius (cm) | Volume (cm³) | Curved area (cm²) | Base area (cm²) | Total area (cm²) |
|---|---|---|---|---|
| 1 | 2.0944 | 6.2832 | 3.1416 | 9.4248 |
| 2 | 16.7552 | 25.1327 | 12.5664 | 37.6991 |
| 3 | 56.5487 | 56.5487 | 28.2743 | 84.8230 |
| 4 | 134.0413 | 100.5310 | 50.2655 | 150.7964 |
| 5 | 261.7994 | 157.0796 | 78.5398 | 235.6194 |
| 10 | 2094.3951 | 628.3185 | 314.1593 | 942.4778 |
| 20 | 16755.1608 | 2513.2741 | 1256.6371 | 3769.9112 |
| 50 | 261799.3878 | 15707.9633 | 7853.9816 | 23561.9449 |
All values calculated with the standard formulas. Volume in cm^3, areas in cm^2.
Frequently asked questions
What is the formula for the volume of a hemisphere?
The volume of a hemisphere is V = (2/3) x pi x r^3, where r is the radius. For a radius of 5 cm that gives (2/3) x 3.14159 x 125 = 261.80 cm^3. If you know the diameter instead, use r = d/2 first, or apply V = (pi x d^3) / 12.
How is hemisphere volume related to sphere volume?
A hemisphere is exactly half a sphere, so its volume is (1/2) x (4/3)pi r^3 = (2/3)pi r^3. To get the full sphere volume just multiply the hemisphere volume by 2. Conversely, to find the hemisphere volume from a known sphere volume, divide by 2.
What is the difference between curved surface area and total surface area?
The curved surface area (A = 2 pi r^2) covers only the dome - the bowl-shaped part. The total surface area (K = 3 pi r^2) adds the flat circular base (B = pi r^2). If you are painting the outside of a dome building you need the curved area; if you are manufacturing a closed bowl you need the total area.
How do I find the radius from the volume of a hemisphere?
Rearrange V = (2/3) pi r^3 to get r = (3V / 2pi)^(1/3). For example, if the volume is 523.6 cm^3, then r = (3 x 523.6 / (2 x pi))^(1/3) = (249.9 / 6.2832)^(1/3) = 39.79^(1/3) = 3.42 cm. This calculator does this automatically when you select "Volume" from the Solve from menu.
How do I convert hemisphere volume between cubic centimetres and cubic inches?
1 cubic inch = 16.3871 cubic centimetres. To convert cm^3 to in^3 divide by 16.3871; to convert in^3 to cm^3 multiply by 16.3871. For example, 261.80 cm^3 / 16.3871 = 15.97 in^3. Switching the unit dropdown in this calculator performs the conversion automatically.
Is a bowl the same as a hemisphere?
A bowl is approximately hemispherical but real bowls have wall thickness and often a flattened base, so the hemisphere formula gives the exterior volume. For the interior capacity, subtract the wall thickness from the outer radius to get the inner radius, then apply the formula to the inner radius. A very thin-walled bowl is close enough to a hemisphere that the formula is a good approximation.
What is the ratio of a hemisphere volume to its total surface area?
V / K = [(2/3) pi r^3] / [3 pi r^2] = r / 3. The ratio equals one-third of the radius, regardless of size. This is a useful consistency check: if you calculate V and K separately, dividing V by K should give you r/3.