Formula and Method for the Volume of a Hemisphere
A hemisphere is exactly half of a sphere, cut through its center. Since a full sphere's volume is V = (4/3)πr³, cutting it in half gives the hemisphere volume formula: V = (2/3)πr³, where r is the radius. This calculator also derives the curved surface area, total surface area, and an optional weight estimate from the same radius.
How the calculation works
Enter the radius and choose its unit. The calculator cubes the radius and multiplies by (2/3)π to get the volume, in cubic units such as ft³ or m³. The curved (lateral) surface — just the dome, like half of a sphere's shell — is A = 2πr², which is half of a full sphere's surface area (4πr²). Adding the flat circular base (πr²) gives the total surface area, A = 3πr². If you enter a material density in kg per cubic meter, the tool converts the volume to cubic meters and multiplies by that density to estimate the hemisphere's weight.
Common mistakes
- Using the sphere formula by mistake: a hemisphere's volume is (2/3)πr³, not the full sphere's (4/3)πr³ — forgetting to halve it doubles the result.
- Radius vs. diameter: the formula needs the radius. If you only know the diameter, divide it by 2 first — using the diameter directly gives a volume roughly 8 times too large.
- Curved area vs. total area: the curved surface area (2πr²) covers only the dome. If the base also needs a material (like paint or a liner), use the total surface area (3πr²) instead.
Real-world applications
- Dome roofs, planetarium domes, and igloo-shaped structures use hemisphere volume and surface area for material and HVAC capacity planning.
- Hemispherical tanks, bowls, and mixing vessels use the volume formula to determine liquid capacity.
- Radar domes (radomes) and satellite dish covers use curved surface area to estimate protective coating or fabric needed.
- Engineering and manufacturing use the weight estimate to plan shipping, structural loads, or material costs for cast or molded hemispherical parts.