Volume of a Hemisphere Calculator
Calculate the volume, curved surface area, total surface area, and circumference of a hemisphere from its radius. See also Volume of Sphere Calculator and Volume of Ellipsoid Calculator.
How to Calculate the Volume of a Hemisphere
A hemisphere is exactly half of a sphere. To find its volume, use the formula V = (2/3)πr³ — which is half the sphere volume formula. The curved surface area covers only the dome, while the total surface area includes the flat circular base. This calculator computes all values from the radius.
Hemisphere Volume Formula
V = (2/3) × π × r³
Curved Surface Area (CSA) = 2 × π × r²
Total Surface Area (TSA) = 3 × π × r²
Circumference = 2 × π × r
Base Area = π × r²
Example
Find the volume of a hemisphere with radius 5:
V = (2/3) × π × r³
V = (2/3) × π × 5³
V = (2/3) × π × 125
V ≈ 261.7994 cubic units
CSA = 2 × π × 25 ≈ 157.0796
TSA = 3 × π × 25 ≈ 235.6194
Hemisphere Volume Reference Table
| Radius | Volume | Curved SA | Total SA |
|---|---|---|---|
| 1 | 2.0944 | 6.2832 | 9.4248 |
| 2 | 16.7552 | 25.1327 | 37.6991 |
| 3 | 56.5487 | 56.5487 | 84.8230 |
| 4 | 134.0413 | 100.5310 | 150.7964 |
| 5 | 261.7994 | 157.0796 | 235.6194 |
| 6 | 452.3893 | 226.1947 | 339.2920 |
| 7 | 718.3775 | 307.8761 | 461.8141 |
| 8 | 1072.3303 | 402.1239 | 603.1858 |
| 9 | 1526.8140 | 508.9380 | 763.4070 |
| 10 | 2094.3951 | 628.3185 | 942.4778 |
| 15 | 7068.5835 | 1413.7167 | 2120.5750 |
| 20 | 16755.1608 | 2513.2741 | 3769.9112 |
| 25 | 32724.9235 | 3926.9908 | 5890.4862 |
| 50 | 261799.3878 | 15707.9633 | 23561.9449 |
| 100 | 2094395.1024 | 62831.8531 | 94247.7796 |
When Do You Need This Calculation?
Hemisphere volume calculations are essential in many practical situations:
- Dome construction: Architects calculate the air volume under hemispherical domes for climate control and acoustics (stadiums, planetariums, observatories).
- Bowl capacity: Manufacturers determine how much liquid or food a hemispherical bowl can hold for product labeling.
- Igloo sizing: Engineers estimate the internal living space of hemispherical shelters.
- Fountain basins: Landscape architects calculate water volume for hemispherical fountain features.
- Half watermelon: Food scientists estimate edible volume of halved spherical fruits.
Solved Examples
Example 1: Dome Stadium Air Volume
A hemispherical dome stadium has an internal radius of 120 m. Calculate the air volume inside.
V = (2/3) × π × 120³
V = (2/3) × π × 1,728,000
V ≈ 3,619,115 m³ (about 3.62 million cubic meters)
Example 2: Hemispherical Bowl
A salad bowl has the shape of a hemisphere with internal radius 12 cm. How many milliliters can it hold?
V = (2/3) × π × 12³
V = (2/3) × π × 1728
V ≈ 3,619.11 cm³ ≈ 3,619 mL ≈ 3.62 liters
Example 3: Igloo Living Space
An igloo is approximately hemispherical with an internal radius of 2 m. What is the livable volume?
V = (2/3) × π × 2³
V = (2/3) × π × 8
V ≈ 16.76 m³
Example 4: Half Watermelon
A spherical watermelon has a diameter of 30 cm. What is the volume of one half?
Radius = 30 / 2 = 15 cm
V = (2/3) × π × 15³
V = (2/3) × π × 3375
V ≈ 7,068.58 cm³ ≈ 7.07 liters
Common Mistakes
Using the sphere formula: A hemisphere is HALF a sphere. Use V = (2/3)πr³, not (4/3)πr³. Using the full sphere formula gives twice the correct answer.
Confusing curved SA with total SA: Curved SA (2πr²) covers only the dome. Total SA (3πr²) includes the flat circular base. Choose the right one for your application.
Using diameter as radius: Remember to halve the diameter. A bowl with diameter 24 cm has radius 12 cm, not 24 cm.
Wrong coefficient: The coefficient is 2/3, not 2/4 or 1/3. It comes from halving the sphere formula (4/3 ÷ 2 = 2/3).
Key Takeaways
- Hemisphere volume = (2/3)πr³ — exactly half the sphere volume.
- Two hemispheres of equal radius combine to form one complete sphere.
- Curved surface area = 2πr²; total surface area = 3πr² (adds the flat base πr²).
- Common real-world hemispheres: domes, bowls, igloos, half fruits.
- To find radius from volume: r = ∛(3V / 2π).
- Doubling the radius increases hemisphere volume by 8× (same scaling as a sphere).
Frequently Asked Questions
What is a hemisphere?
A hemisphere is exactly half of a sphere, cut along a great circle. It has a curved dome surface and a flat circular base. A bowl is a common real-world example of a hemisphere shape.
What is the difference between curved and total surface area?
The curved surface area (CSA = 2πr²) covers only the dome. The total surface area (TSA = 3πr²) adds the flat circular base (πr²). Use TSA when the base is part of the surface, and CSA when only the dome matters.
How does hemisphere volume relate to sphere volume?
The hemisphere volume is exactly half the sphere volume. Sphere V = (4/3)πr³, so hemisphere V = (2/3)πr³. Two hemispheres of the same radius make one complete sphere.
How do I find the radius from the volume?
Rearrange the formula: r = ∛(3V / (2π)). For example, if V = 261.8, then r = ∛(785.4 / 6.2832) = ∛125 = 5.