Radiometric Rock Dating Calculator (K-Ar, U-Pb, Rb-Sr)

Understanding the radiometric rock dating calculator — concept and principle

Radiometric dating finds the age of a rock or mineral from the steady decay of radioactive isotopes inside it. Comparing the daughter isotope built up with the parent that remains dates volcanic rocks, meteorites and Earth's oldest crystals. This calculator supports four classic systems: potassium-argon, two uranium-lead decays and rubidium-strontium.

An Ar/K ratio of 0.5 means about 2.5 half-lives have passed — roughly 3.1 billion years.

How it is calculated

t = (t½ ÷ ln 2) × ln(1 + D/P)

D/P is the measured ratio of radiogenic daughter atoms to remaining parent atoms.
t½ is the half-life of the chosen system: K-40 1.248 billion years, U-238 4.468 billion years, U-235 704 million years, Rb-87 49.7 billion years.
t½ ÷ ln 2 is the mean life, the inverse of the decay constant λ.

For K-Ar, only about 10.72% of K-40 decays produce Ar-40 (most of the rest become Ca-40), so the calculator divides the Ar/K ratio by 0.1072 before using it: t = (1/λ) ln(1 + (Ar/K) ÷ 0.1072). The result is shown in millions of years.

The principle

Each radioactive atom has a fixed probability of decaying per unit time, so the parent decreases exponentially: P = P₀e^(−λt). Every decayed parent becomes a daughter, so D = P₀ − P = P(e^(λt) − 1). Rearranging gives e^(λt) = 1 + D/P, and taking the logarithm gives the formula. The method assumes the sample started with no radiogenic daughter and stayed a closed system, losing or gaining neither parent nor daughter since it formed. Isochron methods are used to test these assumptions.

Worked example

Inputs: D/P (Ar-40/K-40) = 0.5, system K-40 → Ar-40.
Mean life: 1.248 × 10⁹ ÷ 0.693147 = 1.80048 × 10⁹ years.
Branch correction: 0.5 ÷ 0.1072 = 4.66418, so 1 + D/P = 5.66418.
ln(5.66418) = 1.73416.
t = 1.80048 × 10⁹ × 1.73416 = 3,122.33 million years (3,122.329872), about 3.1 billion years.

Each point is the calculator re-run with only “Daughter/parent ratio D/P” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Daughter/parent ratio D/P	Age (million years)
0.125	1,391.598707
0.25	2,167.061114
0.375	2,707.319778
0.5	3,122.329872
0.625	3,459.372219
0.75	3,743.160026
1	4,203.936718

How to read the result

When D/P = 1 in a simple system, exactly one half-life has passed. Each method has a useful range: K-Ar spans young volcanic rocks to billions of years, U-Pb dates the oldest minerals with high precision, and Rb-Sr suits very old rocks because its half-life is long. A result older than Earth, about 4.54 billion years, signals a problem with the sample or the assumptions.

Common mistakes

Using total Ar instead of radiogenic Ar-40 after correcting for atmospheric argon.
Applying the branch correction to systems other than K-Ar, or forgetting it for K-Ar.

Keep in mind: Published decay constants differ slightly between data sets, and real ages need careful laboratory measurement, blank and contamination corrections, and checks of closed-system behaviour. Treat this as an illustration of the method, not a laboratory result.

Frequently asked questions

Why can't carbon-14 date rocks?

Its half-life is only about 5,730 years, so it suits organic material up to tens of thousands of years old.

Why use two uranium decays?

U-238 and U-235 decay to different lead isotopes at different rates. If both ages agree, the sample was most likely a closed system.

Is the parent gone after one half-life?

No. Half remains after one half-life, a quarter after two and an eighth after three.

Sources: G. B. Dalrymple, The Age of the Earth, Stanford University Press, 1991 · G. Faure and T. M. Mensing, Isotopes: Principles and Applications, 3rd ed., Wiley, 2005

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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