What is the volume formula for a hemisphere?

The volume of a hemisphere is two-thirds pi r cubed, which is exactly half the volume of a full sphere (four-thirds pi r cubed). For a radius r, volume = (2/3) times pi times r cubed. This formula applies to a solid hemisphere. For a hollow hemispherical shell, you would subtract the volume of the inner hemisphere from the outer.

What is the difference between curved and total surface area of a hemisphere?

A hemisphere has two surfaces: the curved surface (the bowl-shaped outer surface) and the flat circular base (the cut face). The curved surface area equals 2 pi r squared, which is half the full sphere's surface area of 4 pi r squared. The flat base area is pi r squared. The total surface area is the sum of both: 3 pi r squared.

Where do hemisphere calculations come up in real life?

Hemisphere formulas appear in architecture for dome design, in engineering for pressure vessel end caps, in cookware for bowl capacity, in astronomy for describing half-sphere sky surveys, and in medicine for calculating the volume of brain hemispheres. The flat base area is used when the hemisphere sits on a flat surface and that contact area is needed.

Hemisphere Calculator

A hemisphere is exactly half of a sphere, created by cutting a sphere through its centre. It has a curved outer surface that is half the full sphere's surface area, and a flat circular base where the cut was made. The two surfaces together make up the total surface area of the hemisphere. Hemispheres appear throughout engineering, architecture, and science: geodesic domes are approximately hemispherical, pressure vessel end caps take the hemispherical form because it distributes stress evenly, radar domes (radomes) protect antennae in the familiar dome shape, and igloo designs exploit the hemisphere's structural efficiency. In medicine, brain anatomy divides into left and right hemispheres. Calculating the volume and surface areas from the radius is a common task in Year 11 and 12 maths, engineering design, and 3D printing where you need to estimate material quantities. This calculator takes the radius and returns the volume, the curved surface area (the bowl shape), the flat base area (the circular disk), and the total surface area (curved plus flat). Enter radius in any consistent unit and all four results appear instantly. The default uses r = 6, giving vol = 452.39, curved SA = 226.19, flat SA = 113.10, total SA = 339.29.

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Radius (r)

units

452.39

volume (cubic units)

Curved SA226.19

Flat base SA113.10

Total SA339.29

Diameter12.00

Volume = (2/3)πr³. Curved SA = 2πr². Flat SA = πr². Total SA = 3πr².

How it works

For a hemisphere of radius r: volume = (2/3) * pi * r^3. The curved (bowl) surface area = 2 * pi * r^2. The flat circular base area = pi * r^2. The total surface area = curved SA + flat SA = 3 * pi * r^2. The diameter is 2r. These follow directly from the full sphere formulas (volume = (4/3) * pi * r^3, surface area = 4 * pi * r^2) by halving the volume and curved surface, then adding the circular base.

Worked example

For r = 6: volume = (2/3) * pi * 216 = 144pi = 452.39. Curved SA = 2 * pi * 36 = 72pi = 226.19. Flat SA = pi * 36 = 36pi = 113.10. Total SA = 3 * pi * 36 = 108pi = 339.29. Diameter = 12. These match the defaults pre-filled above.

Related calculators

Torus Volume Calculator: volume and surface area of a donut-shaped solid.
Frustum Calculator: volume and surface area of a truncated cone.
Annulus Calculator: area of a flat ring shape.
Circle Segment Calculator: area and arc of a circular segment.

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