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Nuclear radius questions look like formula substitution, yet students regularly lose marks on them: a cube root taken as a square root, femtometres left out, or a density calculation that does not cancel A. Both the JEE (Main) 2026 syllabus and the NEET (UG)-2026 syllabus list "composition and size of nucleus". This post explains what the formula means, why it makes nuclear density constant, and works four problems. It does not repeat mass defect and binding energy, which we cover in a separate post.
What does R = R0 A^(1/3) say?
The NCERT Class 12 chapter Nuclei states that a nucleus of mass number A has radius R = R0 A^(1/3), where R0 = 1.2 × 10^-15 m, that is 1.2 fm (1 fm = 10^-15 m). The radius is found from scattering experiments. NCERT notes that electron scattering senses the charge distribution while alpha particles sense the nuclear matter, so the two methods give slightly different radii.
The key word is cube root. Doubling A does not double the radius; it multiplies it by 2^(1/3), about 1.26. To double the radius you need eight times the mass number.
Why is nuclear density the same for every nucleus?
Treat the nucleus as a sphere:
Volume V = (4/3)πR^3 = (4/3)π(R0 A^(1/3))^3 = (4/3)π R0^3 A.
Mass is about A atomic mass units: M ≈ A u.
Density ρ = M ÷ V = A u ÷ ((4/3)π R0^3 A) = u ÷ ((4/3)π R0^3).
A cancels. The density does not depend on which nucleus you pick. NCERT compares nuclei to "a drop of liquid of constant density" and gives nuclear density as about 2.3 × 10^17 kg m^-3, against 10^3 kg m^-3 for water. It also notes that neutron stars have comparable density.
Worked example 1: radii of two nuclei
Find the radii of aluminium-27 and tellurium-125.
27^(1/3) = 3, so R(Al) = 1.2 × 3 = 3.6 fm.
125^(1/3) = 5, so R(Te) = 1.2 × 5 = 6.0 fm.
Ratio of radii = 6.0 ÷ 3.6 = 5/3 ≈ 1.67.
Ratio of volumes = (5/3)^3 = 125/27 ≈ 4.63, exactly the ratio of mass numbers, as it must be.
Worked example 2: calculate the density
Use the CODATA value of the atomic mass constant, u = 1.6605 × 10^-27 kg.
R0^3 = (1.2 × 10^-15)^3 = 1.728 × 10^-45 m^3.
(4/3)π × 1.728 × 10^-45 = 4.1888 × 1.728 × 10^-45 ≈ 7.238 × 10^-45 m^3.
ρ = 1.6605 × 10^-27 ÷ 7.238 × 10^-45 ≈ 2.29 × 10^17 kg m^-3.
If you use the proton mass, 1.6726 × 10^-27 kg, you get about 2.31 × 10^17 kg m^-3, which rounds to NCERT's 2.3 × 10^17. Either is accepted if you say which mass you used. NCERT's own Example 13.1 for iron-56 gives 2.29 × 10^17 kg m^-3. Compared with water, the ratio is 2.29 × 10^17 ÷ 10^3 ≈ 2.3 × 10^14.
Worked example 3: run the formula backwards
Which nucleus has half the radius of a nucleus with A = 216?
R is proportional to A^(1/3), so halving R means dividing A by 2^3 = 8.
A = 216 ÷ 8 = 27. Aluminium-27 has half the radius.
Check: 216^(1/3) = 6 and 27^(1/3) = 3, and 3 is half of 6.
Worked example 4: Earth at nuclear density
How small would Earth be if it had nuclear density? The NASA Earth fact sheet gives Earth's mass as 5.9722 × 10^24 kg.
V = M ÷ ρ = 5.9722 × 10^24 ÷ 2.29 × 10^17 ≈ 2.6 × 10^7 m^3.
r = (3V ÷ 4π)^(1/3) = (6.2 × 10^6)^(1/3) ≈ 184 m.
The whole planet would fit inside a sphere about 370 m across. Ordinary matter is mostly empty space between nuclei; nuclear matter is not.
Which mistakes should you avoid?
Square root instead of cube root. R grows as A^(1/3), not A^(1/2).
Using Z. The formula uses mass number A, not atomic number Z.
Forgetting fm. 1.2 fm is 1.2 × 10^-15 m. Convert before calculating density.
Cubing only R0 or only A^(1/3). Cube the whole radius.
Using atomic radius. The formula is for the nucleus only. NCERT's Points to Ponder note that the mass density of the atom does not follow this rule.
A practice routine
Derive "density is independent of A" from memory, as NCERT exercise 13.10 asks, until it takes under a minute.
Learn the perfect cubes 8, 27, 64, 125 and 216; examiners pick mass numbers that use them.
Solve five ratio problems (radius, volume, surface area) without a calculator.
Do one density problem using u and one using the proton mass, and compare.
If picturing why volume tracks mass number is the sticking point, the Fermi Concept Library covers every concept in physics, chemistry, biology, mathematics and computing, each with its own lesson.
Start now: find the radius of a nucleus with A = 64, then the A of a nucleus with twice that radius.
