RMS Voltage Calculator
Enter any voltage measurement and choose your waveform shape to instantly calculate root mean square (RMS) voltage. This calculator supports peak, peak-to-peak, and average voltage inputs across six waveform types: sine, square, triangle, sawtooth, half-wave rectified sine, and full-wave rectified sine. RMS voltage is the DC-equivalent heating value of an AC signal and is what most multimeters display for AC circuits.
What is RMS voltage?
Root mean square (RMS) voltage is the square root of the average of the squared instantaneous voltage values over one complete cycle. It represents the equivalent DC voltage that would deliver the same amount of power to a resistive load. For a 120 V AC mains outlet, 120 V is the RMS value; the peak voltage is actually about 170 V. RMS is the standard AC voltage reported by multimeters and specified on equipment nameplates because power (P = V^2 / R) depends on the square of voltage, so the square-averaging makes physical sense.
How to convert between peak, peak-to-peak, average, and RMS
The conversion factor between peak voltage and RMS depends entirely on the shape of the waveform. For a pure sine wave, Vrms = Vp / sqrt(2), or about 0.7071 times the peak. A square wave at 50% duty cycle has Vrms = Vp because it spends equal time at the positive and negative peak with no smooth transition. A triangle or sawtooth wave gives Vrms = Vp / sqrt(3), approximately 0.5774 times the peak, because the waveform spends more time near zero. Half-wave rectification halves the effective power delivery, giving Vrms = Vp / 2. Use the reference table above for a quick lookup of any waveform.
DC offset and composite RMS
When an AC signal rides on top of a DC level (a DC offset), the total RMS is not simply the sum of the two voltages. Instead, you use the composite RMS formula: Vrms_total = sqrt(Vrms_AC^2 + V0^2), where V0 is the DC offset. This formula falls directly from the definition of RMS because the cross-term averages to zero over a full cycle. A common example is a bipolar switching power supply output that has a ripple voltage superimposed on a DC rail.
Form factor and crest factor explained
Two dimensionless ratios describe the shape of a waveform. The form factor is Vrms divided by the average rectified voltage (Vavg). For a sine wave, the form factor is pi / (2 * sqrt(2)), approximately 1.111. A perfect square wave has a form factor of exactly 1. A higher form factor means the waveform has a more pronounced shape relative to its mean level. The crest factor is the peak voltage divided by the RMS voltage. For a sine wave, this is sqrt(2), about 1.414. High crest factors (such as 2.0 for a half-wave rectified sine) indicate that the peak is much higher than the RMS, which matters for component ratings, insulation, and power quality analysis.
RMS voltage formulas by waveform type
| Waveform | Vrms from Vp | Vrms from Vpp | Vrms from Vavg | Crest factor |
|---|---|---|---|---|
| Sine wave | Vp / sqrt(2) = 0.7071 Vp | Vpp / (2*sqrt(2)) = 0.3536 Vpp | (pi/(2*sqrt(2))) * Vavg = 1.1107 Vavg | sqrt(2) = 1.4142 |
| Square wave (50%) | Vp | Vpp / 2 | Vavg | 1.0000 |
| Triangle wave | Vp / sqrt(3) = 0.5774 Vp | Vpp / (2*sqrt(3)) = 0.2887 Vpp | (pi/(2*sqrt(3))) * Vavg = 0.9069 Vavg | sqrt(3) = 1.7321 |
| Sawtooth wave | Vp / sqrt(3) = 0.5774 Vp | Vpp / (2*sqrt(3)) = 0.2887 Vpp | (pi/(2*sqrt(3))) * Vavg = 0.9069 Vavg | sqrt(3) = 1.7321 |
| Half-wave rectified sine | Vp / 2 = 0.5000 Vp | Vpp / 4 = 0.2500 Vpp | (pi/2) * Vavg = 1.5708 Vavg | 2.0000 |
| Full-wave rectified sine | Vp / sqrt(2) = 0.7071 Vp | Vpp / (2*sqrt(2)) = 0.3536 Vpp | (pi/(2*sqrt(2))) * Vavg = 1.1107 Vavg | sqrt(2) = 1.4142 |
Standard conversion formulas between peak (Vp), peak-to-peak (Vpp), average (Vavg), and RMS voltages. All formulas assume zero DC offset and a symmetric, periodic waveform.
Frequently asked questions
Why is RMS voltage lower than peak voltage?
An AC waveform continuously rises and falls, spending much of its time at voltages below the peak. The RMS calculation squares the instantaneous voltage (making all values positive), averages those squared values over a full cycle, then takes the square root. Because squaring a sinusoid and averaging gives half the peak-squared value, the RMS is the peak divided by sqrt(2), which is about 70.7% of the peak. The waveform is simply not at its maximum the whole time.
What is the RMS voltage of standard household mains electricity?
In North America, the mains supply is 120 V RMS at 60 Hz. The peak voltage is 120 x sqrt(2), approximately 169.7 V. In most of Europe, the supply is 230 V RMS at 50 Hz, with a peak of about 325 V. All voltages printed on plugs, sockets, and appliances are RMS values unless explicitly stated otherwise.
How does a multimeter measure AC voltage?
Most inexpensive multimeters in "AC voltage" mode measure the average rectified voltage and then multiply by the sine-wave form factor (1.111) to display an RMS-calibrated reading. This is accurate only for clean sine waves. True-RMS multimeters sample the instantaneous voltage many times per cycle, square each sample, average them, and take the square root, giving accurate results for any waveform including square waves, triangle waves, and distorted mains signals.
Does a square wave have higher RMS voltage than a sine wave at the same peak?
Yes. A square wave at 50% duty cycle has Vrms = Vp, so all of the peak voltage contributes to the RMS. A sine wave has Vrms = Vp / sqrt(2), which is about 70.7% of the peak. At the same peak voltage, a square wave therefore delivers more power to a resistive load than a sine wave. That is why switching power supplies can be smaller than their linear counterparts for a given power output.
What happens to RMS voltage when I add a DC offset?
A DC offset increases the total RMS voltage. The correct formula is Vrms_total = sqrt(Vrms_AC^2 + V0^2), where V0 is the DC offset. For example, a 10 V peak sine wave (RMS of 7.071 V) with a +5 V DC offset has a total RMS of sqrt(7.071^2 + 5^2) = sqrt(50 + 25) = sqrt(75), approximately 8.66 V. You cannot simply add the two values because the cross-product term cancels over a full cycle.