How this calculator works
- Step 1Small angle2π√(L/g)The textbook period.
- Step 2AmplitudeAGM(1, cos θ₀/2)Corrects for wide swings.
- Step 3ExactT₀ ÷ AGMThe true period of an ideal pendulum.
For small swings a pendulum’s period depends only on its length and gravity. Wider swings take a little longer, and the simple formula stops being accurate.
The calculator gives the exact period using the arithmetic–geometric mean, a fast way to evaluate the elliptic integral behind the full equation.
Worked example: a 1 m pendulum at 10°
- T₀
2π √(L ÷ g)- T
T₀ ÷ AGM(1, cos(θ₀ ÷ 2))
- Small-angle period2π √(1 ÷ 9.8)2.0071 s
- Amplitude correctionAGM(1, cos 5°)+0.19%
- Exact period2.0109 s
Read the result
Released from horizontal (90°), the swing takes 18.03% longer than the small-angle formula says.
- A quarter of the length1.0040 sHalf the period: it goes with √L.
- On the Moon4.9460 sWeaker gravity, slower swing.
Assumptions and limits
- Point massAll mass at the end of a light rod.
- No frictionAir drag and pivot friction are ignored.
- Mass-freeThe mass does not change the period.
- Real bobsA large bob behaves as a physical pendulum.
Related physics tools
A pendulum's period depends on its length and on gravity, not on its mass.
To see how gravity differs between planets, use the weight calculator. The gravitational force calculator applies Newton’s law to any two masses.
Common mistakes
Mistake: Measuring length to the top of the bob.
Instead: Measure to the centre of the bob.
Mistake: Using 2π√(L/g) for wide swings.
Instead: Above about 15°, use the exact period.
Mistake: Timing half a swing as the period.
Instead: The period is there and back.
Checking the method
A separate script simulates the full pendulum equation step by step and matches every exact period, and the known 90° ratio of 1.18034.
Read how we test our calculators, or browse more maths calculators.