What it does
An alternating voltage keeps changing: a sine wave goes up to a peak V₀, down through zero to −V₀ and back, fifty times a second in India. Its plain average over a cycle is zero, yet it clearly heats a kettle. So we need a better “size” for AC.
Heating in a resistor depends on v²/R, and v² is never negative. Take the mean of v² over a cycle, then the square root, and you get the root-mean-square (rms) value. A steady (DC) voltage equal to V_rms would heat the resistor exactly as much. For a sine wave, V_rms = V₀/√2 ≈ 0.707 V₀. That is why the 230 V on the label of Indian mains is an rms value: the voltage actually swings between +325 V and −325 V.
This page draws two cycles of the waveform you choose, its square (shaded), the mean of the square (dashed) and the rms level (green). It adds up the square in 2000 thin slices, takes the square root, and compares it with the exact formula. It also gives the plain mean, the mean of the size |v|, and the power the waveform would deliver into a 100 Ω resistor.
It follows the alternating current chapter of Class 12 Physics (AC voltage applied to a resistor, rms or effective values, power). AC itself comes from a turning coil in faraday induction; to see rms values in circuits with L and C use LCR phasors, and to change AC voltages use the transformer calculator.
How to use it
- Choose a waveform: Sine, Square, Triangle, Sawtooth, Half-wave (only the positive halves of a sine) or Full-wave (the negative halves flipped up).
- Set the peak voltage V₀ with the slider, from 1 to 400 V, or press Indian mains for a 325 V peak sine.
- Read the rms value, the working and the facts.
Worked example
The page opens with Indian mains: a sine wave with a peak of 325 V.
- Squaring every value and averaging over a cycle gives a mean of v² of 52 800 V² (exactly V₀²/2 = 325²/2 = 52 812.5 V²).
- V_rms = √52 812.5 = 230 V, which is 325/√2.
- The mean over a cycle is 0 V, but the mean of the size |v| is 2V₀/π = 207 V.
- Into a 100 Ω resistor it delivers V_rms²/R = 528 W, the same as a steady 230 V.
Switch to Square with the same peak and the rms is the full 325 V; switch to Half-wave and it is 325/2 = 163 V.
Tips
- AC meters read rms values. The ratings on appliances (230 V, 5 A) are rms too.
- Peak = √2 × rms holds only for a sine wave. Other shapes have other ratios.
- Insulation must survive the peak voltage, not the rms, so 230 V wiring is built for well over 325 V.
FAQ
Why not just use the average? For a symmetric wave the average is zero, which tells you nothing about how much it heats or lights things. The rms value does.
Does the frequency change the rms value? No. Only the shape and the peak matter. The time axis is labelled for 50 Hz, but any frequency gives the same rms.
Is anything saved? No. Everything is worked out on this page and nothing leaves your device.
Where the numbers come from
- Root-mean-square value V_rms = √(mean of v²) and the standard results for a sine wave (V₀/√2), square wave (V₀), triangle and sawtooth (V₀/√3), half-wave (V₀/2) and full-wave rectified sine (V₀/√2). Standard mathematics.
- India's nominal supply of 230 V at 50 Hz is the standard nominal mains voltage in India; check your local supplier for exact limits.
- Checked against an independent NumPy script (mean of v² over 10⁶ samples) and the exact integrals.
Standard scientific values, such as constants, atomic weights and melting points, are facts found in every textbook; we name where ours were checked. If you are an author or publisher and want a value, a credit or a table changed or removed, tell us and we will act quickly.
Last updated 2026-11-09.