Radiometric Dating Calculator

Geology-focused isotope dating with timescale suitability guidance.

Worked example — Radiometric Dating Calculator

Estimated age	3.952 kyr
Uncertainty range	2.796 kyr – 5.295 kyr
Half-life	5.73 kyr
Decay constant	1.21 × 10⁻⁴ /y
Typical useful window	0–60 kyr

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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The clock only works if nothing leaked 🖖

Radiometric dating relies on the steady, statistical law of radioactive decay: N(t) = N₀ e^(-λt). In any sample containing billions of unstable parent atoms, the probability of decay per unit time is a constant, unaffected by temperature, pressure, or chemical bonds. This yields a clock of remarkable accuracy. By measuring the ratio of remaining parent isotopes to accumulated daughter isotopes, geologists can calculate the time elapsed since the rock cooled and locked its crystal structure. However, this clock requires a closed system. If parent or daughter isotopes escape due to weathering or metamorphism, the clock is reset or distorted. Modern geochronologists use multiple decay systems (like U-Pb and K-Ar) in parallel to cross-verify ages and detect open-system behavior.

Half-life: nature's built-in stopwatch 🖖

The key idea is the half-life: the fixed time it takes for half of a radioactive parent isotope to decay into its daughter. After one half-life 50% of the parent remains; after two, 25%; after three, 12.5%. Because the fraction halves each interval, the amount of parent left tells you how many half-lives have passed. Enter a parent fraction of 0.25 and the tool reports an age of two half-lives — for carbon-14 (half-life ≈ 5,730 years) about 11,460 years.

The bomb pulse dates the living 🖖

Atmospheric nuclear weapons tests between 1955 and 1963 nearly doubled the amount of carbon-14 in the air, creating a sharp bomb pulse. After the 1963 test-ban treaty this excess radiocarbon fell off steadily as it mixed into oceans and plants. Forensic scientists now read that decline in tooth enamel or eye-lens tissue to pin a person's year of birth to within a year or two — and to unmask forged wines and paintings made after 1950.

RADIOMETRIC DATING — PICKING A CLOCK THAT MATCHES THE SAMPLE

Which Isotope Clock Fits Your Sample?

Every radiometric clock runs the same equation, t = ln(1 + D/P)/λ, and every one of them is useless outside its own window. A half-life too short and the parent is gone; too long and no measurable daughter has built up. So the choice is made twice over: by how old you think the sample is, and by what the material actually contains.

Carbon-14 — organic material, the last 60 000 years 5730 y — to ~60 kyr

Potassium-argon — volcanic rock, hundreds of thousands to billions of years 1.25 × 10⁹ y

Uranium-lead — zircon crystals, the deep past 4.47 × 10⁹ y

Rubidium-strontium — very old rock, when the daughter was already there 4.88 × 10¹⁰ y

Carbon-14 — organic material, the last 60 000 years

What you know: The sample was once alive: wood, charcoal, bone, textile. Dating is by the fraction of ¹⁴C still present, not by a daughter product.

Half-life: 5730 y — to ~60 kyr

Worked example: 62% of the original ¹⁴C remains, half-life 5730 y → t = ln(1/0.62)/λ ≈ 3950 years

Open this case: wood artifact

Carbon-14 — organic material, the last 60 000 years. A short half-life buys precision on human timescales and nothing beyond them. The sample was once alive: wood, charcoal, bone, textile. Dating is by the fraction of ¹⁴C still present, not by a daughter product. — A short half-life buys precision on human timescales and nothing beyond them.

Potassium-argon — volcanic rock, hundreds of thousands to billions of years

What you know: The sample is an igneous rock that crystallised from a melt. The daughter is argon gas, trapped in the mineral only after it solidified.

Half-life: 1.25 × 10⁹ y

Worked example: Ar/K = 1.2 with a half-life of 1.248 × 10⁹ y → t = ln(2.2)/λ ≈ 1.42 billion years

Open this case: basalt flow

The clock starts when the melt solidifies and traps the first argon atom.

Uranium-lead — zircon crystals, the deep past

What you know: The sample contains zircon. The crystal accepts uranium as it grows but rejects lead, so any lead inside it was made by decay in place.

Half-life: 4.47 × 10⁹ y

Worked example: Pb/U = 0.98 with a half-life of 4.468 × 10⁹ y → t = ln(1.98)/λ ≈ 4.40 billion years

Open this case: zircon crystal

Uranium-lead — zircon crystals, the deep past. Uranium in, lead out: a crystal that keeps two clocks and lets you check one against the other. The sample contains zircon. The crystal accepts uranium as it grows but rejects lead, so any lead inside it was made by decay in place. — Uranium in, lead out: a crystal that keeps two clocks and lets you check one against the other.

Rubidium-strontium — very old rock, when the daughter was already there

What you know: The half-life is 48.8 billion years, ten times the age of the Earth. Some strontium-87 was present from the start, so a single measurement is not enough.

Half-life: 4.88 × 10¹⁰ y

Worked example: Sr/Rb = 0.05 → t = ln(1.05)/λ ≈ 3.4 billion years, and the honest version needs several minerals to pin down the initial strontium

Open the U-Pb case and switch the system to Rb-Sr

Rubidium-strontium — very old rock, when the daughter was already there. A daughter that was already present: one measurement is not enough, so geologists fit a line through several. The half-life is 48.8 billion years, ten times the age of the Earth. Some strontium-87 was present from the start, so a single measurement is not enough. — A daughter that was already present: one measurement is not enough, so geologists fit a line through several.

References (1)

Insight block 3 — the bomb pulse dating living tissue: K. L. Spalding, B. A. Buchholz, L.-E. Bergman, H. Druid and J. Frisén, "Age written in teeth by nuclear tests." Nature 437, 333–334, 2005.

Problem solved in full

A wood sample that retains 62% of its original carbon-14 5 steps
A wood sample retains 62% of its original carbon-14. Date it — then find both ends of the method: where it runs out of signal, and why the age is less precise than the measurement behind it.
1. Decay is exponential, and the half-life converts into a decay constant through ln 2. A half-life of 5730 years becomes 1.21 × 10⁻⁴ per year.
  
  $N = N_{0} e^{- λ t}, λ = \frac{ln 2}{t _{1/2}} = \frac{0.6931}{5730} = 1.21 \times 1 0^{- 4} y r^{- 1}$
2. Invert the decay law for time. Taking logarithms turns the surviving fraction straight into an age.
  
  $t = \frac{1}{λ} ln \frac{1}{f} = \frac{ln ( 1/0.62 )}{1.21 \times 1 0 ^{- 4}} = \frac{0.4780}{1.21 \times 1 0 ^{- 4}} = 3952 yr$
3. Check it against the half-life rather than trusting the arithmetic: 3952 years is 0.690 half-lives, and 2^(−0.690) is 0.620. The two routes agree, as they must, since ln 2 was the only bridge between them.
  
  $\frac{3952}{5730} = 0.690 half-lives, 2^{- 0.690} = 0.620 ✓$
4. Now differentiate. The relative error in the age is the relative error in the fraction divided by ln(1/f) — a factor of 2.09 here. A measurement good to 1% yields an age good to 2.1%, and the amplification worsens for younger samples, because ln(1/f) shrinks towards zero as f approaches 1.
  
  $\frac{Δ t}{t} = - \frac{1}{ln ( 1/ f )} \cdot \frac{Δ f}{f} = - 2.09 \frac{Δ f}{f}$
5. At the other end the signal runs out. After ten half-lives — 57 300 years — 0.098% of the original carbon-14 is left, which is where counting statistics lose to the background.
  
  $10 t_{1/2} = 57300 yr : 2^{- 10} = 0.098% remains$
Answer

The tool prints 3.952 kyr from a 5.73 kyr half-life and a decay constant of 1.21 × 10⁻⁴ per year. The two things it cannot print are the walls on either side. Young samples fail by error amplification: f near 1 makes ln(1/f) near 0 and the age uncertainty blows up, which is why nobody carbon-dates something from the last century. Old ones fail by exhaustion — a thousandth of the original atoms after ten half-lives — which is why the useful window shown on the panel stops near 60 kyr. Every dating method has this pair of walls and only the numbers between them differ. Switch the system to U–Pb: the half-life jumps by nearly six orders of magnitude, both walls move with it, and the method that just dated a campfire cannot touch anything younger than a million years.

$t = \frac{ln ( 1/ f )}{λ} = 3952 yr; \frac{Δ t}{t} = \frac{1}{ln ( 1/ f )} \cdot \frac{Δ f}{f}$

Learning path

Clocks in the rock

Leads to Molecular clock the same law solved for time instead of for the amount left.

By the 42Math team · Updated 2026-07-27

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

wood artifact - Carbon-14 dating for organic remains in archaeological timescales.
basalt flow - K-Ar style volcanic rock age estimate in million-year range.
zircon crystal - U-Pb zircon dating for deep geological time.