Hemisphere Calculator
Calculate volume, curved surface area, and total surface area of any hemisphere.
What is a Hemisphere?
A hemisphere is exactly half of a sphere, created by cutting a sphere through its centre along a great circle. The resulting solid has two surfaces: the curved dome (half the sphere’s surface) and the flat circular base (the cross-section disc). Together, these define the total surface area, while the enclosed three-dimensional space gives the volume.
The word “hemisphere” comes from the Greek words for half (hemi) and sphere (sphaira). In geography, the Earth is commonly divided into the Northern and Southern Hemispheres along the equator, and the Eastern and Western Hemispheres along the prime meridian. In mathematics and engineering, however, a hemisphere refers specifically to the geometric half-sphere shape described by the formulas below.
The volume of a hemisphere is (2/3) pi r cubed, exactly half the volume of the full sphere (4/3) pi r cubed. The curved surface area is 2 pi r squared, which is also exactly half the sphere’s surface area of 4 pi r squared. The total surface area adds the flat circular base (pi r squared) to give 3 pi r squared. These clean relationships make the hemisphere one of the more elegant three-dimensional shapes to work with mathematically.
In the physical world, hemispheres appear as bowls, domes, igloo structures, the caps of cylindrical tanks, satellite dish reflectors, and the cross-sectional ends of pressurised vessels. The hemisphere is optimal for certain engineering applications because the dome shape efficiently distributes compressive loads, which is why many ancient and modern large-span structures use hemispherical or near-hemispherical roofs.
A remarkable result from Cavalieri’s principle connects the hemisphere to simpler shapes: a hemisphere of radius r has the same volume as a cylinder of radius r and height r with a cone of the same dimensions removed from it. This elegant relationship - (pi r cubed) minus ((1/3) pi r cubed) = (2/3) pi r cubed - was one of Archimedes’ celebrated discoveries.
Formula and Derivation
Given radius r:
Given volume V:
How to Use This Calculator
- Choose your input - select “Enter Radius” if you know the radius, or “Enter Volume” if you know the volume and want to work backwards to find the radius and surface areas.
- Type the value - for radius, use a length unit (cm, m, inches, feet). For volume, use cubic units matching the length unit (cm cubed, m cubed, etc.).
- Click Calculate - the calculator shows all four measurements: radius, volume, curved surface area, and total surface area.
- Understand curved vs total SA - the formula note explains the difference: curved SA is the dome only, while total SA adds the flat base circle.
- Try worked examples - click any “Try this example” link below to pre-fill the calculator with real values and see the full calculation.
Example Calculations
Example 1 - Radius of 5 units (bowl)
A hemispherical bowl has a radius of 5 cm. Find its volume (capacity) and surface areas.
Example 2 - Radius of 10 units (dome structure)
An architectural dome has a radius of 10 m. Find the volume of enclosed space and the roof area.
Example 3 - Volume of 2094 cubic units
A tank holds 2094 litres (cubic decimetres). The tank is hemispherical. Find the radius and surface areas.
Example 4 - Radius of 3 units
A small plastic hemisphere (like a bowl lid) has radius 3 cm. Find its dimensions.