The carbon dating calculator above converts the amount of carbon-14 left in a sample into a radiocarbon age. You can enter the result as a percent modern carbon figure, as a specific activity in disintegrations per minute per gram of carbon alongside the modern reference value, or as an age you want to invert back into a carbon-14 content. It handles both half-life conventions, propagates a measurement uncertainty into an age uncertainty, and reports the mean life it used so the arithmetic can be checked.
Arb Digital publishes free calculators that are explicit about the difference between a computed number and a usable answer. That distinction matters more here than almost anywhere else on the site: the number this page produces is a conventional radiocarbon age, and a conventional radiocarbon age is not a calendar date. Converting one to the other requires a calibration curve, and that step is outside what any single formula can do.
What This Carbon Dating Calculator Does
Carbon-14 is produced continuously in the upper atmosphere and enters living tissue through photosynthesis and the food chain. While an organism is alive its carbon-14 content stays in equilibrium with the atmosphere. Once it dies, exchange stops and the carbon-14 decays with nothing replacing it. Measuring how much is left, relative to a modern standard, therefore measures how long ago exchange stopped. The Oxford Radiocarbon Accelerator Unit sets out the mechanism and the decay relation on its page explaining how radiocarbon dating works.
This tool does the arithmetic of that conversion and nothing beyond it. It takes the remaining fraction, applies exponential decay, and returns years before present, where "present" is fixed at 1950 by convention. It also shows what the same measurement would give on the other half-life, so you can see the size of the difference for yourself rather than taking it on trust.
A boundary worth stating: our half-life calculator works the general problem in either direction for any nuclide, most often asking how much is left after a given time. This page does the specific inverse — how much time has passed given what is left — with the carbon-14 half-life and the 1950 datum built in, plus the reporting conventions that come with radiocarbon specifically.
How to Use It
- Choose your input. Percent modern carbon if that is what the laboratory reported; activity if you have counts per gram; an age if you want to run the calculation backwards.
- Leave the half-life on Libby unless you have a specific reason to change it. Conventional ages are reported on 5,568 years, and calibration curves expect that.
- Enter an uncertainty in pMC to see the corresponding age uncertainty. It grows as the sample gets older, because the same absolute error covers more years further down the decay curve.
- Read the mean life in the third result box. Multiplying the negative natural log of the remaining fraction by that number is the whole calculation.
- Treat the date as uncalibrated. The fourth box gives the raw year implied by the age, which is not the calendar date of the sample.
The Formula and How It Is Calculated
Radioactive decay is first order, so the remaining fraction f follows f = e−t/τ, where τ is the mean life. Solving for time gives t = −τ ln f. The mean life is the half-life divided by the natural logarithm of 2: for the Libby half-life of 5,568 years that is 8,033 years, and for the Cambridge half-life of 5,730 years it is 8,267 years. The ORAU dating page writes the decay relation directly in that form, using 8033 as the constant.
Check the default. A sample at 50 percent modern carbon has f = 0.5, so ln f is −0.693147, and multiplying by −8,033 gives 5,568 years — exactly one Libby half-life, as it must. Switch to the Cambridge half-life and the same measurement gives 5,730 years, about three percent older.
Uncertainty propagates through the logarithm rather than linearly. Differentiating gives σt = τ σf / f, so an absolute error in the measured fraction translates into a larger age error the smaller f becomes. Half a pMC on a modern-ish sample is worth a few decades; the same half a pMC on a sample at 2 pMC is worth two centuries. This is the quantitative reason radiocarbon runs out of resolution beyond about 50,000 years: there is so little carbon-14 left that the counting error swamps the signal. Our logarithm calculator and exponential growth calculator handle the underlying maths in general form.
Why the Wrong Half-Life Is Still the Right One to Report
Libby's original measurement of the carbon-14 half-life gave 5,568 years. Later work showed the true value is closer to 5,730 years, so the number the whole field standardised on is known to be about three percent wrong. It is still the number used for reporting, and that is deliberate rather than inertia.
ORAU explains the reasoning on its page on radiocarbon calibration: a conventional radiocarbon age is calculated on the assumption that atmospheric radiocarbon concentration has always matched its 1950 value and that the half-life is 5,568 years, and both of those assumptions are corrected together at the calibration stage. Because calibration curves are constructed from measurements reported the same way, switching half-life at the laboratory stage would break the correction rather than improve it.
The practical rule follows directly. Report conventional radiocarbon ages on the Libby half-life, then calibrate. Never apply the Cambridge half-life and then feed the result into a calibration curve, because the curve has already absorbed that offset. The Cambridge option on this page exists so you can see the difference, not so you can substitute it into a dating workflow.
Why an Uncalibrated Age Is Not a Date
The second assumption behind a conventional age is that atmospheric carbon-14 has been constant. It has not. Production varies with solar activity and the Earth's magnetic field, and the oceans exchange carbon on long timescales, so the atmospheric concentration has drifted by several percent over the Holocene and considerably more further back. ORAU puts the consequence plainly: a raw figure does not mean the sample comes from the year that simple subtraction implies, both because atmospheric radiocarbon has varied by a few percent over time and because the true half-life differs from the value used.
Calibration fixes this by comparing radiocarbon measurements against material of independently known age — tree rings counted one by one, then corals, speleothems and varved sediments further back. The resulting curve is not a smooth line. It has plateaux where the atmospheric concentration happened to stay flat for a century or two, and a radiocarbon age landing on a plateau calibrates to a wide, sometimes multi-peaked range of calendar dates rather than a single year. That is why published radiocarbon dates come as probability ranges rather than as points.
Two more corrections sit alongside calibration in real work. Isotopic fractionation means different tissues take up carbon-13 and carbon-14 at slightly different rates, so laboratories normalise measurements to a standard carbon-13 value before computing an age. And reservoir effects matter for anything that took its carbon from the ocean or from groundwater rather than the atmosphere: marine organisms can appear several hundred years older than contemporaneous land material because deep water carries older carbon. None of these are handled here, and all of them are handled by the reporting laboratory.
What Radiocarbon Can and Cannot Date
The method only works on material that once exchanged carbon with the atmosphere or the biosphere: wood, charcoal, bone collagen, seeds, textiles, parchment, shell. Stone tools cannot be dated, only organic material found in association with them, which is why the archaeological argument about what a date actually dates is often more contested than the measurement itself.
The upper limit is set by counting statistics rather than by the physics. After ten half-lives roughly one part in a thousand of the original carbon-14 remains, and separating that from contamination and instrumental background becomes the dominant problem. Practical limits sit around 50,000 years, and a very small amount of modern contamination — a fingerprint, a rootlet, a conservation treatment — makes an ancient sample look far younger than it is. One percent of modern carbon added to an infinitely old sample produces an apparent age of about 37,000 years.
The lower end has its own complication. Atmospheric nuclear testing in the 1950s and 1960s nearly doubled atmospheric carbon-14, creating the "bomb pulse", and fossil fuel burning dilutes it in the other direction. Material from the last seventy years can therefore read above 100 pMC, which this calculator will report as a negative age because that is the honest arithmetic answer. Post-1950 material is dated against the bomb-pulse curve rather than by ordinary decay. For handling the counting statistics behind any of this, the standard deviation calculator and the significant figures calculator are useful, and very small fractions are easier to read through the scientific notation converter.
Arb Digital publishes hundreds of free calculators across chemistry, physics, maths and finance — no sign-up, no limits. If something you need is missing, tell us and we will look at building it.
Browse All Free Tools Suggest a ToolCommon Mistakes to Avoid
- Treating the result as a calendar date — a conventional radiocarbon age must go through a calibration curve before it corresponds to years in the ordinary sense.
- Using the Cambridge half-life and then calibrating — calibration curves are built for the Libby convention, so substituting the better half-life first double-corrects the result.
- Forgetting that "present" means 1950 — years BP are counted from 1950, not from today, so a 2,000 BP sample is not from two thousand years before this year.
- Assuming uncertainty is constant — the same absolute error in measured carbon-14 covers far more years in an old sample, because the error propagates through a logarithm.
- Ignoring reservoir effects — marine and freshwater organisms take up older carbon and can appear centuries older than land material of the same age.
Related Free Tools From Arb Digital
Work the general decay problem in either direction with the half-life calculator, handle exponential relationships in general with the exponential growth calculator, and evaluate logs directly using the logarithm calculator. Summarise counting scatter with the standard deviation calculator, keep reported precision honest with the significant figures calculator, and read very small fractions with the scientific notation converter. The full free online tools hub lists everything else.
Frequently Asked Questions
Living things exchange carbon with the atmosphere and hold a carbon-14 content matching it. Exchange stops at death and the carbon-14 decays without replacement, so measuring how much remains relative to a modern standard measures how long ago that exchange stopped.
Age equals minus the mean life multiplied by the natural logarithm of the remaining fraction. The mean life is the half-life divided by the natural log of 2, which is 8,033 years for the Libby half-life and 8,267 years for the Cambridge value.
Libby's 5,568 years, for reporting. It is known to be about three percent low, but conventional radiocarbon ages are defined on it and calibration curves are built to match, so substituting the more accurate value before calibrating introduces an error rather than removing one.
Years before present, where present is fixed at 1950 by convention. The datum was chosen because it precedes widespread atmospheric nuclear testing, which sharply raised atmospheric carbon-14 from the mid-1950s onwards.
Because atmospheric carbon-14 has not been constant. Production varies with solar activity and the geomagnetic field, so a raw age must be compared against material of independently known age, such as counted tree rings, to become a calendar range.
Practically around 50,000 years. Beyond that so little carbon-14 remains that contamination and instrument background dominate the measurement, and a tiny amount of modern carbon makes a very old sample appear far younger.
Because the sample reads above 100 percent modern carbon. Atmospheric testing in the 1950s and 1960s nearly doubled atmospheric carbon-14, so recent material can exceed the 1950 standard and is dated against the bomb-pulse curve rather than by ordinary decay.
This calculator is provided for education and general reference. It computes an uncalibrated conventional radiocarbon age and is not a laboratory, archaeological or dating service; calibrated dates and their uncertainties should come from an accredited radiocarbon laboratory.
