Half-Life Calculator

Radioactive decay and any exponential half-life process — solve for remaining quantity, elapsed time, or the half-life itself

Half-life describes exponential decay — the time it takes for a quantity to reduce by half — and while radioactive decay is the classic example, the same mathematical pattern governs any process that decreases by a consistent proportion over consistent time intervals, which is why this calculator solves for remaining quantity, elapsed time, or the half-life itself depending on what's known.

Why exponential decay isn't the same as linear decay

A quantity undergoing exponential decay loses a fixed percentage of its current amount each half-life period, not a fixed absolute amount — which means the absolute quantity lost keeps shrinking each period even though the proportional loss (50% per half-life, by definition) stays constant. This is fundamentally different from linear decay, where a fixed absolute amount is lost each period regardless of how much remains. The three variables — starting quantity, remaining quantity, elapsed time, and half-life itself — are all mathematically related through the exponential decay formula, and given any three of them, the fourth can be solved for directly, which is exactly what lets this calculator work in multiple directions depending on which value is actually the unknown one.

A worked example

A quantity that halves every fixed time period loses much more in absolute terms during its first half-life than during its fifth half-life, even though each period represents an identical 50% proportional reduction — this is the defining signature of exponential versus linear decay, and it's why extrapolating decay using a linear "loses X per period" assumption produces an increasingly wrong answer the further out the extrapolation goes.

How this connects to your other science and math calculations

For understanding the underlying exponential and logarithmic relationships this calculator's formula depends on, the Exponent Calculator and Log Calculator cover the more general mathematical operations that half-life calculations are built from.

Common mistakes

Assuming a quantity undergoing half-life decay loses a constant absolute amount per time period (linear thinking) rather than a constant percentage (exponential thinking) leads to significant errors, especially over multiple half-life periods — the gap between a linear extrapolation and the true exponential result grows larger the more half-life periods have elapsed.