Half-Life Calculator

SIMPLIFIED

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Fill in all three, then press =.

What the top and bottom each do

The unit is yours — measure in hours and the half-life comes out in hours.

Decay constant h

How long until half of it is gone. A rate or the constant k work just as well — pick one above.

Give all three and the half-life comes out.

Amount at the start P

How much there was to begin with — the dose, the mass, the value.

Amount left A

How much was still there when you measured. It has to be less than the start.

Time elapsed t

How long the decay ran for. This must be in the same unit as the half-life — if the half-life is in years, so is this.

Measure how much is left and the half-life follows — that is how a sample is dated. It also reads straight off the constant k or a percentage rate, and gives you all three whichever you start from.

Examples — tap one and it solves

How it works out, step by step

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A lab does not know the half-life. It knows what it started with, what is left, and how long it waited — and the half-life comes out of those three.

That is the direction this half-life calculator works in — a half life calculator that starts from what you measured, not from an answer you already have. It also converts from the decay constant k or a percentage rate, and gives you all three whichever one you begin with.

What a Half-Life Is

The time it takes for half of something to disappear. Ten grams becomes five, five becomes 2.5, and so on.

The part that surprises people is that it does not depend on how much you have. Ten grams and ten tonnes of the same isotope share it. A single atom has no schedule at all — it decays when it decays — but across billions of atoms the average is fixed, and that average is the half-life.

Which is why it is such a useful number. It belongs to the substance, not to the sample.

The Half-Life Formula

Two lines cover everything.

The half life formula comes in two lines. From the decay constant: h = ln 2 ÷ k. And backwards, k = ln 2 ÷ h. The ln 2 is there because you are asking how long until half is left — a different fraction would put a different logarithm in its place.

From a measurement: first k = −ln(A/P) ÷ t, then the line above. Put together, h = ln 2 × t ÷ ln(P/A).

A logarithm is unavoidable here, and it is worth seeing why. The half-life sits in the exponent of A = P(½)^t/h, and no amount of ordinary algebra brings it down. Logarithms exist precisely to do that.

How to Find the Half-Life From a Measurement

You need three numbers, and a lab has all three.

Say 100 units drop to 25 in 10 hours. A quarter is left, which is two halvings, so it is 5 hours — and you can see that without any formula. The calculator is for when it is not a neat fraction.

Carbon dating is exactly this sum. A living thing keeps its carbon-14 topped up; once it dies, the topping up stops and the clock starts. Measure what is left against what a living sample has, and the 5,730-year half-life converts that into an age. 29.83% left means about 10,000 years.

The same arithmetic dates rocks with uranium, sets the gap between doses of a drug, and decides how long radioactive waste has to be stored. Different fields, one calculation.

How to Use This Half-Life Calculator

Pick what you have at the top. From a measurement asks for the start, what is left, and the time. From k and from a rate each want a single number, and the other boxes disappear.

The unit is yours entirely. Measure in hours and it comes out in hours; measure in millennia and it comes out in millennia. There is nothing to convert, because the same unit goes in and out.

One tap shows it alongside k and the equivalent percentage rate, plus the working step by step. If you already know the half-life and want to know how much will be left, that is the exponential decay calculator; for things going up instead of down, the exponential growth calculator gives the doubling time, which is this same formula with the sign reversed. It is free to use online, with no account.

Half-Life FAQ

What is the half-life formula?

h = ln(2) divided by k, where k is the decay constant. If you are working from a measurement instead, k = -ln(A/P) divided by t, so the half-life is ln(2) x t divided by ln(P/A).

How do you find the half-life from data?

Measure how much you started with, how much is left, and how long it took. The calculator does the rest — it is one logarithm, because the half-life sits in the exponent and nothing else brings it down.

How is the half-life related to the decay constant?

They are two ways of saying the same thing: h = ln(2)/k and k = ln(2)/h. A large k means fast decay and a short half-life. Physics tends to use k, while chemistry and medicine use the half-life.

What is the half-life of carbon-14?

5,730 years. That is what makes radiocarbon dating work: measure how much carbon-14 is left in a sample, and the half-life tells you how long it has been decaying.

Does the half-life depend on how much you start with?

No, and that is what makes it useful. Ten grams and ten tonnes of the same substance have the same half-life. Only the rate matters, never the quantity.

Start, remainder, time — and one logarithm. The half-life belongs to the substance, so once you have it, it is yours for every sample of the same thing.