Example 1
Reference example: one full cycle lasts 2 s. A frequency of 0.5 Hz is 30 cycles per minute, and ten cycles last 20 s.
f = 0.5 Hz
Frequency from Period Calculator
Enter the period T, the time of one full cycle, in seconds. The calculator divides 1 by that time with f = 1/T and reports the frequency in hertz, together with the cycles per minute and the frequency at half the period. For T = 2 s the result is 0.5 Hz, and for T = 0.02 s it is 50 Hz. The model needs a steady, positive period of one cycle; it does not compute wavelength or angular frequency ω.
Result
Enter the period of one cycle in seconds. The frequency, the cycles per minute, and the f(T) chart will appear here.
The reverse formula T = 1/f is on the oscillation period calculator. Angular velocity from the same period is on the ω = 2π/T calculator. Frequency from a spring constant and a mass is on the spring calculator.
How it works
1one cycle
÷ T2 s
f0.5 Hz
Frequency says how many full cycles fit into a unit of time. If one cycle lasts a time T, then for a motion with a constant period f = 1/T. The unit is the hertz: 1 Hz = 1 cycle per second. Formally the hertz is the special name of the reciprocal second for periodic phenomena. That definition is in every textbook on oscillations (Halliday, Resnick, Walker; Young and Freedman) and in the SI Brochure.
The calculator uses one number: the period T in seconds. It divides 1 by that number. For T = 2 s the result is f = 0.5 Hz, which is 30 cycles per minute. Ten such cycles last 20 s. At half the period, T = 1 s, the frequency is 1 Hz, because f is inversely proportional to T. A period of 1/60 s gives 60 Hz, the US mains frequency. A period of 0.02 s gives 50 Hz.
What the result panel shows. Besides f in hertz, the calculator reports the cycles per minute (60/T), the entered period in a convenient unit, the frequency at half the period, and the time of ten cycles (10·T). The chart draws the hyperbola f(T) from ½T to 4T. An empty field shows no result. T = 0 and a negative number end in a message, because a physical period is a positive time.
Limits of the model. The formula describes a train of cycles with one constant period. When the period changes, the result is the reciprocal of the entered T, not an instantaneous frequency. The calculator does not find angular velocity (that is the ω = 2π/T calculator), the period from a frequency (T = 1/f), a pendulum period from a length, or a spring frequency from k and m. It also does not find a wavelength: that also needs the wave speed.
The unit switch in the header does not change seconds or hertz, because they are shared by both systems. It changes only the decimal separator and the grouping of digits. The values in this text stay in SI units. Convert milliseconds before you enter them: 20 ms is 0.02 s.
Examples on the page: T = 2 s → 0.5 Hz; the mean of ten cycles, T = 2.01 s → 0.4975 Hz; a US mains cycle, 1/60 s → 60 Hz; a period of 0.0022727 s → 440 Hz; a metronome every 1 s → 1 Hz; a sway of 4 s → 0.25 Hz; a swell of 10 s → 0.1 Hz; a solar day of 86,400 s → 1.157·10⁻⁵ Hz; a resonator period of 1/32,768 s → 32,768 Hz. The button “Fill this example” enters the same number and refreshes the result panel and the chart.
Formula
f = 1 / T
[f] = Hz = 1/s for periodic phenomena. The inverse is T = 1/f. The link to angular velocity, computed on a separate page, is ω = 2πf.
Model: the definition of frequency as the reciprocal of the period (kinematics of oscillation). The hertz is named after Heinrich Hertz; the 11th CGPM adopted it in 1960. This calculator finds only f. Order of operations: 1/T in seconds, then 60/T, 2/T, and 10·T.
Theory, derivation, and domain
Definition of frequency
Frequency f is the number of repetitions of a periodic phenomenon in a unit of time. If N full cycles occur in a time t, then f = N/t. When you count one cycle, the time t is the period T and the formula shortens to f = 1/T. The SI unit is the hertz. The SI Brochure defines the hertz as the special name of the unit s⁻¹, used for periodic phenomena so that it stays distinct from other quantities with the same dimension, such as angular velocity.
Period and frequency describe the same train of cycles from two sides. A long period is a low frequency. A short period is a high frequency. The conversion is exact: f returns to T by T = 1/f, and T returns to f by the formula on this page. Neither formula needs a mass, a string length, or a spring constant. Those inputs matter only when the period is not yet known.
The hertz: a unit, not the author of the formula
The formula f = 1/T has no single author. It is a definition, used since people began to count the vibrations of strings, pendulums, and waves. The unit is named after Heinrich Hertz, who studied electromagnetic waves. The SI adopted the hertz in 1960, at the 11th General Conference on Weights and Measures, in place of the older name “cycle per second.” The name on the unit does not mean that Hertz derived the reciprocal of the period.
In symbols, 1 Hz = 1/s. In physics prose the word hertz stays with cycles, and it is not mixed with the radian per second. Two hertz means two full cycles per second. The same numeral 2 on an angular velocity would mean 2 rad/s, which is a different description of the motion.
Derivation of f = 1/T
Step 1. Pick an event that repeats: the return of a pendulum to the same extreme, one full sweep of the mains voltage, one vibration of a resonator. The time between successive events is the period T. Step 2. The number of cycles in one second is the number of times T fits into 1 s, so f = 1/T. Step 3. In one minute there are 60 times as many cycles: n = 60/T = 60·f. Step 4. The time of ten cycles is 10·T, which is a stopwatch check of the measurement.
Step 5. For T = 2 s: f = 1/2 = 0.5 Hz, n = 30 cycles per minute, and 10 cycles last 20 s. Half that period is 1 s and f = 1 Hz. Step 6. For T = 1/60 s: f = 60 Hz. For T = 0.02 s: f = 50 Hz. For T = 0.0022727 s, a rounding of 1/440, the panel shows 440 Hz at four significant figures.
Assumptions: one full cycle and a constant period
The field takes the time of one full cycle, not of half a cycle. On a pendulum, the trip to the other side lasts T/2. Entering half the period doubles the frequency relative to the full cycle. When a stopwatch covered several cycles, divide the time by their number first. A sum of 20.1 s over 10 cycles is T = 2.01 s and f = 0.4975 Hz, not the result of entering 20.1 s.
The formula is exact when successive periods are equal. For an oscillation that speeds up or slows down, 1/T is the reciprocal of that one measured cycle. Instantaneous frequency needs the time derivative of the phase. The calculator does not estimate it from one field.
Frequency, the hertz, and angular velocity
These three notations are easy to confuse, because all of them can start from the same T. Frequency f = 1/T is in hertz and counts full cycles. Angular velocity ω = 2π/T is in radians per second and counts angle. For T = 2 s the frequency is 0.5 Hz, while the angular velocity is about 3.14 rad/s. Those are not two writings of the same number. This calculator leaves ω on the angular-velocity page and reports only f as the primary result.
In the harmonic-motion description x(t) = A·cos(2πft) the frequency in hertz stands in the argument next to 2π. The same motion is also written A·cos(ωt), where ω = 2πf. Both writings agree. This page computes the first of the two: f from T. It does not compute the amplitude A.
Orders of magnitude when T is the given
The range starts with cycles a stopwatch can see. A swell with a period of 10 s has f = 0.1 Hz. A sway with a period of 4 s has 0.25 Hz. A metronome that clicks every second has 1 Hz and 60 clicks per minute. A full cycle of 2 s has 0.5 Hz. A solar day treated as one cycle, T = 86,400 s, has a frequency of 1.157·10⁻⁵ Hz. That is the frequency of the day’s repetition, not the angular velocity of the Earth’s rotation.
On the short-period side: 1/60 s is 60 Hz, 0.02 s is 50 Hz, and 0.0022727 s returns 440 Hz. A watch crystal is built for 32,768 Hz, so one cycle lasts 1/32,768 s and the calculator returns 32,768 Hz. Visible light has frequencies of hundreds of terahertz. That period is not measured with a stopwatch; this page does not convert a wavelength into hertz.
Sources
The definition, the unit, and the distinction between the hertz and other quantities of dimension 1/s come from the SI Brochure (BIPM) and from oscillations textbooks: D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, chapters on harmonic motion; H. D. Young and R. A. Freedman, University Physics, the oscillations chapter. The link between f and ω in a harmonic oscillator is in R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. I, Chapter 21. ISO 16 sets A4 at 440 Hz. This page is educational.
How to use
- Enter the period T in seconds, the time of one full cycle, for example 2.
- If a stopwatch covered several cycles, divide the time by their number. Do not enter a sum of 20.1 s over 10 cycles as the period.
- Convert milliseconds to seconds: 20 ms is 0.02. A comma and a period are the same decimal separator.
- Click Calculate or wait for the live result. For T = 2 s the calculator shows 0.5 Hz.
- Read the panel: cycles per minute, the entered period, the frequency at half the period, and the time of ten cycles. Clear removes the data and hides the panel.
Symbol glossary
f is the frequency, and T is the period of one full cycle. For T = 2 s the result is 0.5 Hz.
- f
- Frequency [Hz], f = 1/T. The number of full cycles per second.
- T
- Period [s], the time of one full cycle. It must be greater than zero.
- Hz
- Hertz, 1 Hz = 1/s for periodic phenomena. Named after Heinrich Hertz; an SI unit since 1960.
- n
- Cycles per minute, n = 60/T = 60·f. This is not an angular velocity.
- ω
- Angular velocity [rad/s], ω = 2πf = 2π/T. This calculator does not report it as the result.
Worked examples
Example 2
A stopwatch covered 10 cycles in 20.1 s. Enter the mean period, 2.01 s, not the sum. The result is 0.4975 Hz, about half a hertz, not 2 Hz.
f = 0.4975 Hz
Example 3
One cycle of US mains power lasts 1/60 s. The calculator shows 60 Hz, which is 3,600 cycles per minute. The period on the panel reads 0.01667 s.
f = 60 Hz
Example 4
A tuning fork set to 440 Hz has a period of 1/440 s. The entered rounding 0.0022727 s shows 440 Hz on the panel.
f = 440 Hz
Example 5
A metronome clicks every 1 s. That is 1 Hz and 60 clicks per minute. At half that period the frequency is 2 Hz.
f = 1 Hz
Example 6
A measured sway period is 4 s. The frequency of that cycle is 0.25 Hz, which is 15 cycles per minute.
f = 0.25 Hz
Example 7
A swell with a period of 10 s has a frequency of 0.1 Hz. Six such cycles fit in one minute.
f = 0.1 Hz
Example 8
A solar day treated as one cycle lasts 86,400 s. The frequency of that repetition is 1.157·10⁻⁵ Hz. This is not the angular velocity of the Earth.
f = 1.157·10⁻⁵ Hz
Example 9
A watch crystal is built for 32,768 cycles per second. The period of one cycle is 1/32,768 s, and the calculator returns 32,768 Hz.
f = 32,768 Hz
Common questions
What does this calculator compute, and from which formula?
The calculator finds the frequency from f = 1/T: it divides 1 by the period of one full cycle. For T = 2 s the result is 0.5 Hz, which is 30 cycles per minute. Ten such cycles last 20 s, and at half the period the frequency is 1 Hz.
Which unit should I use for the period?
Seconds. Divide milliseconds by 1,000: 20 ms is 0.02 s, and 1 divided by 0.02 s is 50 Hz. Multiply minutes by 60. A comma and a period are the same decimal separator. The unit switch in the header does not change seconds or hertz, because they are shared by both systems; it changes only how numbers are written.
What if a stopwatch timed several cycles, not one?
Divide the measured time by the number of cycles and enter the mean period. A sum of 20.1 s over 10 cycles is T = 2.01 s and f = 0.4975 Hz. Entered 20.1 s would be read as one long period.
Why does a period of 2 s not give a frequency of 2 Hz?
Because frequency is the reciprocal of the period, not a copy of it. Two hertz requires T = 0.5 s. A period of 2 s is half a cycle per second, that is 0.5 Hz. A longer period means a lower frequency.
How does frequency differ from angular velocity?
Frequency f counts full cycles per second and uses the hertz. Angular velocity ω counts radians per second and equals 2πf. This calculator finds only f. The ω = 2π/T calculator finds the angular velocity from the same period.
Where do I compute the period when I already know the frequency?
On the oscillation-period calculator, with T = 1/f. This page does the reverse: T is the input and f is the result. The two formulas are reciprocals of each other.
What does this calculator not compute?
It does not compute a pendulum period from a length, a spring frequency from k and m, a wavelength, or a wave speed. Separate calculators do that. Here the only input is a period you already measured or were given.
Why is there no result for T = 0 or for a negative number?
Division by zero is undefined, and a zero period would mean infinitely many cycles per second. A physical period is a positive time. The calculator requires T > 0 and until then shows a message instead of a result.
Does the formula work when the period changes?
The formula f = 1/T assumes a constant period. For an oscillation that speeds up or slows down, the result is the reciprocal of the entered time of one cycle, not an instantaneous frequency. Instantaneous frequency is the time derivative of the phase.
What is a hertz, and where does the name come from?
The hertz is the SI unit of frequency: 1 Hz = 1 cycle per second, formally the reciprocal second for periodic phenomena. The name comes from Heinrich Hertz. The formula f = 1/T is the definition of frequency, not a law named after Hertz. The 11th CGPM adopted the hertz in 1960.
How should I measure the period so the frequency is reliable?
Time many full cycles and divide by their number. That reduces the reaction-time error. Enter the mean period of one cycle, not the sum. The trip of a pendulum to the other side is T/2, not the full period.
Can I recover 440 Hz from the period of a tuning fork?
Yes, if you enter the time of one cycle. Rounding 1/440 to 0.0022727 s shows 440 Hz on the panel at four significant figures. The calculator does not read a note name. It requires the time of one cycle in seconds.
How do I calculate frequency with f = 1/T?
Divide 1 by the period of one cycle in seconds. The result is in hertz. For T = 2 s: f = 1/2 = 0.5 Hz.
What is f when T = 2 s?
0.5 Hz. In one minute that is 30 cycles, because 0.5·60 = 30.
What is f when T = 0.02 s?
50 Hz. That is mains frequency in much of Europe: one cycle lasts 20 ms.
How do I convert milliseconds to seconds before calculating?
Divide by 1000. A period of 20 ms is 0.02 s; f = 1/0.02 = 50 Hz. Entering 20 instead of 0.02 inflates the result a thousandfold.
Does the calculator find wavelength or angular frequency ω?
No. The result is f = 1/T. Wavelength needs a wave speed, and ω = 2πf belongs on the angular-velocity page.
Sources
The definition, the unit, and the derivation below come from established physics and metrology sources.
- Wikipedia — Frequency
- Wikipedia — Hertz (the unit, 11th CGPM, 1960)
- BIPM — The International System of Units (SI), hertz as the name of s⁻¹
- Feynman Lectures on Physics, Vol. I, Ch. 21 — The Harmonic Oscillator
- NIST — Metric (SI) program
- Halliday, D.; Resnick, R.; Walker, J., Fundamentals of Physics — harmonic motion, f = 1/T.
- Young, H. D.; Freedman, R. A., University Physics — Oscillations: period and frequency.
- ISO 16, standard musical pitch A4 = 440 Hz.
Page updated November 3, 2026.