# Half-Life Calculator

> Solve exponential decay: enter three of initial amount, remaining amount, half-life and time elapsed, and find the fourth plus the decay constant.

Category: Mathematics · Applies to: Worldwide · Last updated: 2026-10-29
Interactive calculator: https://calculators.geekmidst.com/calculators/half-life-calculator

Fill in any three of the four boxes and leave the one you want blank. Each half-life halves what is left, whatever the substance.

Also called: Radioactive decay calculator, Exponential decay calculator.

## Inputs
- Initial quantity (N₀) (default 100; optional)
- Remaining quantity (N) (default ; optional)
- Half-life (T½) (default 5730; optional)
- Time elapsed (t) (default 11460; optional)

## Results
- Answer
- Fraction remaining
- Number of half-lives elapsed
- Initial quantity
- Remaining quantity
- Half-life
- Time elapsed
- Decay constant λ (per time unit)
- Mean lifetime τ

## Formula

```
N = N₀ × (½)^(t ÷ T½);  T½ = t × ln 2 ÷ ln(N₀ ÷ N);  t = T½ × log₂(N₀ ÷ N);  λ = ln 2 ÷ T½;  τ = 1 ÷ λ
```

Where:
- N₀: starting quantity
- N: quantity remaining after time t
- T½: half-life
- λ: decay constant
- τ: mean lifetime

## Worked example
1. t ÷ T½ = 11,460 ÷ 5,730 = 2 half-lives.
2. N = 100 × (½)² = 100 × ¼ = 25.
3. λ = ln 2 ÷ 5,730 ≈ 0.000120968 per year.

**Result:** After 11,460 years, two carbon-14 half-lives, 25 of the original 100 units remain.

## What the result means

After 7 half-lives less than 1% remains, and after 10 about 0.1%, which is why “ten half-lives” is a common rule of thumb for “essentially gone”.

## Assumptions and limits
- Simple first-order (exponential) decay with a constant half-life.
- All quantities use the same unit, and the half-life and time use the same time unit.
- Real samples can contain several isotopes or decay chains, and drug elimination often departs from a single half-life; those need more detailed models.

## Questions

### Does the amount ever reach zero?

In the model it never quite does: each half-life halves what is left. In practice a sample becomes undetectable, or for atoms, the last one decays at a random moment.

### How do I find a half-life from two measurements?

Enter the starting and remaining amounts and the time between them, and leave the half-life blank. For 100 → 12.5 in 30 hours, that is three halvings, so T½ = 10 hours.

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Source: GM¹X CalcVerse — https://calculators.geekmidst.com/calculators/half-life-calculator