Quick answer
N = N₀ × (½)^(t ÷ T). 100 units with a half-life of 5 after 15 time units leave 100 × (½)³ = 12.5, or 12.5% of the original. The calculator can also solve for time, half-life or initial amount.
Choose what to solve for, then enter the other values. The calculator returns the result and the fraction remaining.
Related tools: Half Birthday Calculator: The date six months after a birthday. Half Angle Calculator: Half-angle identities evaluated at an angle. Dew Point Calculator: Dew point from temperature and relative humidity.
How to use the Half-Life Calculator
- Choose solve for: Remaining amount, Elapsed time and Half-life.
- Enter initial amount (above 0).
- Optionally, enter remaining amount (above 0).
- Optionally, enter elapsed time (above 0).
- Optionally, enter half-life (above 0).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Solve for | Choose one: Remaining amount; Elapsed time; Half-life | Starts at Remaining amount |
| Initial amount | A number | > 0 |
| Remaining amount (optional) | A number | > 0 |
| Elapsed time (optional) | A number | > 0 |
| Half-life (optional) | A number | > 0 |
Formula and method
Formula
N = N₀ × (½)^(t ÷ T); t = T log₂(N₀ ÷ N); T = t ÷ log₂(N₀ ÷ N)
IEEE-754 double precision; shown to the decimals indicated.
Use the same time unit for elapsed time and half-life.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Solve for = Remaining amount; Initial amount = 100; Elapsed time = 15; Half-life = 5.
| Result | Value |
|---|---|
| Remaining amount | 12.5 |
| Fraction remaining | 12.5% |
Example 2
Inputs: Solve for = Elapsed time; Initial amount = 100; Remaining amount = 25; Half-life = 8.
| Result | Value |
|---|---|
| Elapsed time | 16 |
| Fraction remaining | 25% |
Example 3
Inputs: Solve for = Half-life; Initial amount = 80; Remaining amount = 10; Elapsed time = 12.
| Result | Value |
|---|---|
| Half-life | 4 |
| Fraction remaining | 12.5% |
The first line is the headline result (remaining amount). Below it the half-Life Calculator lists fraction remaining, which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Solving for time
t = T × log₂(N₀ ÷ N). To go from 100 to 12.5 with a half-life of 5: log₂(8) = 3, so t = 15.
Decay constant
The half-life T and decay constant λ are related by T = ln 2 ÷ λ ≈ 0.693 ÷ λ, and the mean lifetime is 1 ÷ λ. Competitor calculators offer these conversions; this page works with the half-life directly.
How the result changes with initial amount
Holding every other input at the first worked example, this table changes only initial amount. It shows how sensitive the answer is to that one value; each row was produced by running this calculator.
| Initial amount | Remaining amount | Fraction remaining |
|---|---|---|
| 50 | 6.25 | 12.5% |
| 100 | 12.5 | 12.5% |
| 200 | 25 | 12.5% |
| 400 | 50 | 12.5% |
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Elapsed time: left empty shows "Enter the elapsed time." beside that field and no result.
- Remaining amount: entering "150" shows "The remaining amount must be below the initial amount." beside that field and no result.
Choosing between the Half-Life Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Half Birthday Calculator](/lifestyle/half-birthday/) | date of birth | Half birthday |
| [Half Angle Calculator](/math/half-angle-calculator/) | angle and unit | sin(θ/2) |
| [Dew Point Calculator](/science/dew-point/) | air temperature and relative humidity | Dew point |
| [Acceleration Calculator](/science/acceleration-calculator/) | initial velocity, final velocity and time | Acceleration |
| [Mole Calculator](/science/mole/) | start from and molar mass | Amount |
Assumptions and limitations
- Remaining must be below the initial amount
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for elapsed time and remaining amount to confirm this.
Frequently asked questions
What is a half-life?
The time for half of a sample to decay.
After 10 half-lives how much is left?
About 0.098% remains, because (1/2)^10 is 1/1024.
Does it apply to drugs?
Yes, to first-order elimination.
What does the Half-Life Calculator calculate?
Remaining amount, elapsed time or half-life from exponential decay. It returns Remaining amount and Fraction remaining.
What formula does the Half-Life Calculator use?
N = N₀ × (½)^(t ÷ T); t = T log₂(N₀ ÷ N); T = t ÷ log₂(N₀ ÷ N). Use the same time unit for elapsed time and half-life.
What do I need to enter in the half-Life Calculator?
You enter Solve for and Initial amount. Optional: Remaining amount, Elapsed time and Half-life.
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. Remaining must be below the initial amount.
What does the “Solve for” option change?
It sets how the half-Life is worked out. The choices are Remaining amount, Elapsed time and Half-life.
What is an example of a half-Life result?
With solve for Remaining amount, initial amount 100, elapsed time 15 and half-life 5, the remaining amount is 12.5.
Related calculators
- Half Birthday Calculator — The date six months after a birthday.
- Half Angle Calculator — Half-angle identities evaluated at an angle.
- Dew Point Calculator — Dew point from temperature and relative humidity.
- Acceleration Calculator — Average acceleration from change in velocity over time.
- Mole Calculator — Convert between mass, moles and number of particles.
- Density Calculator — Solve mass, density or volume.
Sources
- OpenStax, University Physics Volume 1 (Rice University): mechanics
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.