Nuclear Radius Calculator
Calculate the radius of any atomic nucleus using r = r₀ × A^(1/3) where A is the mass number.
Also gives nuclear volume, density, and surface area.
Atomic nuclei are remarkably uniform in density. This leads to a simple empirical formula for nuclear radius:
r = r0 * A^(1/3)
A is the mass number (protons + neutrons). r0 = 1.2 femtometers (fm) = 1.2 x 10^-15 m is the empirical nuclear radius constant, determined from electron scattering experiments. Some sources use r0 = 1.25 fm for slightly different fits.
The cube-root dependence means nuclear volume scales linearly with A: V = (4/3)pir^3 = (4/3)pi(r0)^3 * A
Since volume is proportional to A and A is proportional to total nucleon mass, the nuclear matter density is approximately constant for all nuclei: rho ≈ m_nucleon / ((4/3)pi(r0)^3) ≈ 2.3 x 10^17 kg/m^3
That density is extraordinary: a teaspoon of nuclear matter would weigh about 2 billion tonnes. A neutron star is essentially a giant nucleus held together by gravity, with similar density throughout.
Worked examples:
- Hydrogen-1 (A=1): r = 1.2 fm (just a proton)
- Carbon-12 (A=12): r = 1.2 * 12^(1/3) = 1.2 * 2.289 = 2.75 fm
- Iron-56 (A=56): r = 1.2 * 56^(1/3) = 1.2 * 3.826 = 4.59 fm
- Uranium-238 (A=238): r = 1.2 * 238^(1/3) = 1.2 * 6.20 = 7.44 fm
The formula breaks down for very light nuclei (A < 4) where the liquid drop model underlying it does not apply well.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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