Math
Exponential Decay Calculator
Model anything that loses the same fraction each period: radioactive samples, medication in the bloodstream, a cooling cup of coffee, or a depreciating car.
Choose the percent, continuous, or half-life form, solve for the missing variable, and get the half-life, a table of values, and a graph.
The amount at t = 0: mass, concentration, value, temperature difference.
Enter 12 for a 12% decrease each period.
Elapsed periods, in the same unit as the rate or half-life.
Only labels the answer. The rate must be per one of these units.
Try an example
Result
Solved value
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- Remaining amount
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- Growth factor per period
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- Rate per period (%)
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- Continuous rate k
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- Half-life
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- Equation
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Student quick launch
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Study path
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What exponential decay means
Exponential decay multiplies the amount by the same factor between 0 and 1 each period. A 12% decay rate means each period keeps 88% of the previous amount. The amount never reaches zero, but it halves over and over at a fixed interval called the half-life.
Three forms of the same curve
| Form | Decay factor per period | Half-life |
|---|---|---|
| Percent: y = a(1 - r)^t | 1 - r | ln(1/2) / ln(1 - r) |
| Continuous: y = a·e^(-λt) | e^(-λ) | ln 2 / λ |
| Half-life: y = a(1/2)^(t/h) | (1/2)^(1/h) | h |
The calculator converts between them, so you can enter a percent rate and read off the half-life, or enter a half-life and read off the decay constant λ = ln 2 / h.
Rearranged formulas
How to use this calculator
- Pick the form that matches your problem: percent per period, continuous with λ, or half-life.
- Choose the unknown. The calculator hides that field.
- Enter the known values. Rates are positive numbers: 12 means a 12% decrease.
- Match the time unit to the rate (a per-day rate needs t in days).
- Read the answer, the half-life, the remaining fraction, and the step-by-step work.
How to read the answer
- The decay factor is what remains each period: 0.88 means 88% remains and 12% is lost.
- Half-life depends only on the rate, not on the starting amount.
- After n half-lives the fraction remaining is (1/2)^n: 1/2, 1/4, 1/8, ...
- Percent change over the whole time is negative and less extreme than r × t because each period removes a fraction of a smaller amount.
Common mistakes
- Entering the decay rate as a negative number here. This page already subtracts it; a negative entry models growth and triggers a warning.
- Solving for time when the remaining amount is larger than the initial amount: decay cannot increase an amount.
- Using a per-year half-life with time in months.
- Rounding the decay factor early, which shifts the answer noticeably after many periods.
Worked examples
Percent decay
500 mg losing 12% per hour for 3 hours
Solved value
340.736
Carbon-14 half-life
100 g of carbon-14 (half-life 5,730 years) after 10,000 years
Solved value
29.829244
Find the half-life
80 g decays to 10 g in 18 days
Solved value
6
Time to reach a target
How long for 1,000 to fall to 250 at 10% per year?
Solved value
13.157627
Find the decay rate
A $1,200 phone is worth $300 after 4 years
Solved value
29.289322
Continuous decay
Solve for time: 100 → 50 with λ = ln 2 / 8
Solved value
8
Work backward
What initial amount leaves 40 after 5 periods at 20%?
Solved value
122.070312
Impossible: amount increased
No half-life exists when the amount grew
Solved value
—
Frequently asked questions
What is the exponential decay formula?+
y = a(1 - r)^t, where a is the initial amount, r the decay rate per period as a decimal, and t the elapsed periods. Equivalent forms are y = a·e^(-λt) and y = a(1/2)^(t/h) with half-life h.
How do I find the half-life from a decay rate?+
Half-life = ln(1/2) / ln(1 - r). For a 12% rate per hour, ln 0.5 / ln 0.88 ≈ 5.42 hours. For a continuous constant λ, half-life = ln 2 / λ.
How do I find the decay rate from two amounts?+
r = 1 - (y / a)^(1/t). If 1,200 becomes 300 in 4 years, (300/1200)^(1/4) = 0.7071, so r ≈ 29.29% per year.
Does exponential decay ever reach zero?+
No. Each period multiplies by a fraction, so the amount shrinks toward zero but never reaches it in the model. In practice it becomes negligible after several half-lives.
What is the difference between decay rate and decay constant?+
The decay rate r is the fraction lost per period in the discrete model. The decay constant λ is the rate in the continuous model e^(-λt). They are linked by 1 - r = e^(-λ).
Can I use this for depreciation?+
Yes. Declining-balance depreciation is exponential decay: a car losing 15% of its value each year follows y = a(0.85)^t.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026