Hemisphere Volume Calculator

A hemisphere volume calculator that computes every key measurement of a half-sphere in one click — volume, curved surface area, flat base area, total surface area — and shows full step-by-step working so you can check the arithmetic yourself. It also runs in reverse: supply a known volume and the tool solves for the radius via the cube-root of the rearranged formula.

Hemispheres appear throughout engineering, architecture, science and everyday life: bowls, domes, tanks, planets, lenses and even cell nuclei are all modelled as hemispheres. Getting the geometry right matters whether you are calculating how much concrete a dome needs, how many litres a hemispherical tank holds, or simply checking your textbook answer.

The formula

A sphere of radius r has volume (4/3) x pi x r^3. A hemisphere is exactly half, so:

V = (2/3) x pi x r^3

For surface areas the hemisphere has two parts:

• Curved (dome) surface area: 2 x pi x r^2
• Flat base (circle): pi x r^2
• Total surface area: 3 x pi x r^2

The calculator uses all four expressions with full floating-point precision (pi taken to fifteen significant figures) and rounds the display to four decimal places.

How it works

Enter the radius in any of the seven supported units (mm, cm, m, km, in, ft, yd). The tool evaluates the three distinct formulas — volume, curved area and flat base area — and sums the two area components for the total. It also derives the diameter from 2r so you can read off the span of the hemisphere directly.

For the reverse calculation (radius from volume), the tool rearranges the volume formula algebraically:

r = cube-root(3V / (2 x pi))

JavaScript’s Math.cbrt is used to evaluate the cube root to full IEEE-754 double precision, which corresponds to about fifteen significant decimal digits — more than enough for any practical application.

Worked example

Suppose you have a hemispherical garden water feature with a radius of 0.4 m. How much water does it hold, and how much render does the dome exterior need?

1. r = 0.4 m
2. r^3 = 0.064 m^3
3. V = (2/3) x pi x 0.064 = 0.1340 m^3 (134 litres)
4. Curved SA = 2 x pi x 0.16 = 1.0053 m^2
5. Flat base = pi x 0.16 = 0.5027 m^2
6. Total SA = 1.5080 m^2

You would need roughly 134 litres of water to fill it and just over 1 m^2 of material to coat the exterior dome. Try entering r = 0.4 m in the calculator above to confirm these figures and see the working laid out line by line.

Formula note

The 2/3 coefficient in the volume formula is exact; it does not come from an approximation. It follows directly from integrating the cross-sectional area of a sphere from 0 to r using the disk method: the integral of pi x (r^2 - x^2) from 0 to r evaluates to pi x (r^3 - r^3/3) = (2/3) x pi x r^3. The surface area of the curved portion is derived from the surface-of-revolution integral, which gives 2 x pi x r^2 — identical in form to half the full sphere’s surface area 4 x pi x r^2.