Exponential Decay Calculator

SIMPLIFIED

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Fill in two of the three, then press =.

What the top and bottom each do

Whatever unit the half-life is in, the time must match it — years with years, hours with hours.

Half-life h

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

Now fill in any two of these three — the third is worked out.

Amount at the start P

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

Amount left A

How much remains. Leave this empty to find it, or fill it in to find how long it took.

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.

Decay questions usually give a half-life, not a rate — so start there. A percentage rate and the continuous constant k describe the very same curve, and each one gives the other two.

Examples — tap one and it solves

How it works out, step by step

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Carbon-14 has a half-life of 5,730 years. Start with 100 units and after 10,000 years there are 29.83 left — and that single number is how radiocarbon dating works.

Decay questions almost never hand you a rate. They hand you a half life, so this exponential decay calculator takes one directly. A percentage rate and the continuous constant k work too, and it converts between all three.

The Exponential Decay Formula

There are three versions in circulation, and they describe exactly the same curve.

From a half-life. A = P × (½)^t/h. Divide the time by the half-life to count how many half-lives have passed, then halve that many times. This is the one worksheets use.

From a percentage rate. A = P(1 + r)^t with r negative. Losing 10% a period means r = −0.1, so the factor is 0.9.

Continuously. A = Pe^−kt. The continuous exponential decay model is what physics uses, because atoms do not wait for the end of a period before decaying.

They are linked by two short lines: k = ln 2 / h, and h = ln 2 / k. Give the calculator any one and it works out the other two, so a question set in one language can be answered in another.

How to Calculate Exponential Decay

Count half-lives. That is the whole method, and it is faster than reaching for the formula.

A drug with a six-hour half-life, 500 mg taken now: after 6 hours 250, after 12 hours 125, after 18 hours 62.5, after 24 hours 31.25. Four half-lives, four halvings, no arithmetic beyond dividing by two.

The formula is for when the time is not a neat multiple. 10,000 years is 1.745 half-lives of carbon-14, and (½)^1.745 = 0.2983 — which is where the 29.83 came from.

To go the other way and find the time, you need a logarithm: t = −ln(A/P) ÷ k. That is the only way to bring the exponent down, and it is exactly what dating a sample requires — measure how much is left, and read the age off the curve.

Why It Never Reaches Zero

Each half-life takes away half of whatever is still there. Half of something is never nothing.

1 half-life	50% left
2 half-lives	25% left
3 half-lives	12.5% left
10 half-lives	about 0.1% left
20 half-lives	about 0.0001% left

In practice ten half-lives is treated as gone — that is the rule for clearing a drug from the body, and for storing radioactive waste. Mathematically there is always a little left, which is why the curve flattens but never touches the axis.

How to Use This Exponential Decay Calculator

Pick what you have at the top — half-life, rate, or the constant k — then fill in two of the three boxes below and press =. The empty one is worked out, and the answer appears in it in blue.

Whatever unit the half-life is in, the time must match: years with years, hours with hours. Each box has an information button that says so.

One tap opens all three values together, the working step by step, and the same decay written in all three forms. If the amount is going up rather than down, the exponential growth calculator handles that direction. It is free to use online, with no account.

Exponential Decay FAQ

How do you calculate exponential decay?

Use A = P x (1/2) to the power t/h, where h is the half-life. Divide the time by the half-life to see how many half-lives have passed, then halve the starting amount that many times.

What is the exponential decay formula?

There are three, and they are the same curve. With a half-life: A = P(1/2)^(t/h). With a percentage rate: A = P(1+r)^t, r negative. Continuously: A = Pe^(-kt). Give any one and the other two follow.

How do you find the half-life from a rate?

Half-life = ln(2) divided by k, where k is the decay constant. From a percentage rate r, k is minus the natural log of (1+r). A rate of minus 10% a period gives a half-life of about 6.58 periods.

What is continuous exponential decay?

Decay that happens smoothly rather than in steps. The continuous exponential decay model is A = Pe^(-kt), where k is the decay constant. It is what physics and chemistry use, because atoms do not wait for the end of a period to decay.

Does exponential decay ever reach zero?

No. Each half-life removes half of whatever is still there, so something always remains. After ten half-lives about 0.1% is left, after twenty about 0.0001% — small, but never nothing.

Half-life, rate, or k — the same curve three ways. Count the half-lives when you can, and reach for the logarithm when you need the time.