RMS Voltage

In one sentence: RMS voltage is the steady voltage that heats a resistor as much as an alternating voltage does, and for a sine wave you get it by dividing the peak voltage by the square root of 2.

$V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}}$

Read it out loud: “V R M S equals V peak divided by the square root of 2. The letters R M S are said one at a time, and they stand for root mean square. V peak is the highest voltage the wave reaches, in volts. The square root of 2 is about 1.414.”

Before you start

The ideas and signs this page uses, explained from zero. For powers of ten, Greek letters and other notation in general, see Math basics.

Alternating voltage and the sine wave: An alternating voltage rises to a positive peak, falls through zero to a negative peak, and comes back, many times every second. Its usual shape is the sine wave, a smooth wave named after the sine function, one of the basic functions of trigonometry. Mains electricity is close to a sine wave, so the rms and peak rules on this page apply to it.
Peak and peak-to-peak voltage: The peak voltage is the largest value the wave reaches, measured from zero. The peak-to-peak voltage is the distance from the most negative point to the most positive point. For a wave that is symmetric about zero, the peak-to-peak value is twice the peak. The two are different numbers, so check which one a label or a meter is giving.
The average, and a bar over a letter, like $\overline{v^2}$: The average of a wave over one full cycle is found by adding up its values over the cycle and dividing by the length of the cycle. For a sine wave the positive and negative halves cancel, so the plain average is zero. A bar written over a quantity, such as $\overline{v^2}$ , means the average of that quantity over one cycle. So $\overline{v^2}$ is the average of the squared voltage.
Square roots and $\sqrt{2}$: The square root of 2, written $\sqrt{2}$ , is the number that, multiplied by itself, gives 2. It is about 1.41421, because $1.41421 \times 1.41421 \approx 2$ . The square root of one half is $\frac{1}{\sqrt{2}}$ , which is about 0.707. Dividing by $\sqrt{2}$ is the same as multiplying by $\frac{1}{\sqrt{2}}$ .
Power, watts and heating: Power is the rate at which energy is used, measured in watts (W), where one watt is one joule per second. A resistor turns electrical power into heat, and the power it turns into heat depends on the voltage squared. A 1000 W heater uses 1000 joules of energy every second.
Hertz and mains supplies: The frequency of mains electricity is 60 Hz in the US and 50 Hz in much of Europe. In each cycle the voltage rises, falls, reverses and returns to where it started. So the wave repeats 50 or 60 times every second, and each full cycle takes about one fiftieth or one sixtieth of a second.

What each symbol means

Symbol	Meaning
$V_{\text{rms}}$	The RMS voltage, the root mean square value of an alternating voltage, measured in volts (V). It is the steady DC voltage that would heat a resistor by the same amount. A standard US outlet is labeled 120 V rms, and a European outlet is labeled 230 V rms.
$V_{\text{peak}}$	The peak voltage, the highest value the wave reaches, measured in volts (V). For a sine wave it is higher than the rms value by a factor of $\sqrt{2}$ , about 1.414. A 120 V rms supply has a peak of about 170 V.
$\sqrt{2}$	The square root of 2, about 1.41421, a plain number with no unit. For a sine wave, dividing the peak by it gives the rms value.

The formula piece by piece

$V_{\text{peak}}$: The peak voltage in volts, the highest value the sine wave reaches. It is the starting point, because the rms value is found by scaling this peak down.
$\sqrt{2}$: The square root of 2, about 1.414. It comes from averaging the square of a sine wave, which gives one half of the peak squared, and the square root of one half is one over the square root of 2.
$\frac{1}{\sqrt{2}}$: One over the square root of 2, about 0.707. Multiplying the peak voltage by 0.707 gives the rms voltage, so for a sine wave the rms value is about 70.7 percent of the peak.
$V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}}$: The whole formula in one sentence: the rms voltage equals the peak voltage divided by the square root of 2, which is the same as multiplying the peak voltage by 0.707.
$V_{\text{rms}} = \sqrt{\overline{v^2}}$: The general definition, which works for any waveform. Square the voltage at every moment, average those squares over one cycle, and then take the square root. For a sine wave this gives the peak divided by the square root of 2.

What the formula is doing, and why

Mains electricity is alternating: the voltage rises, falls, goes negative and repeats many times a second. A single number is still needed to say how strong it is. The input is the peak voltage of the wave, $V_{\text{peak}}$ , and the output is the rms voltage, $V_{\text{rms}}$ , in volts. The rms value is the steady voltage that gives the same heating in a resistor.

Why not the average? The average of a sine wave over a full cycle is zero, because the positive and negative halves cancel. But heating depends on the voltage squared, since the power in a resistor is $V^2 / R$ , and squaring makes every part positive. So the useful average is taken after squaring. The name gives the steps in order: square, take the mean, then take the root. That is root mean square.

Why divide by $\sqrt{2}$ ? The square of a sine wave, averaged over a cycle, comes out to one half of the peak voltage squared. Taking the square root gives $V_{\text{rms}} = V_{\text{peak}} \times \sqrt{\frac{1}{2}} = \frac{V_{\text{peak}}}{\sqrt{2}}$ . So the rms value is about 0.707 times the peak, or about 70.7 percent of it.

Why mains is quoted in rms. Labels give the rms value because it matches the heating and the power a device gets from the supply. A 120 V US supply has a peak of about 170 V, and a 230 V European supply has a peak of about 325 V. Insulation and components must withstand the peak, so their ratings must be above the peak, not only above the rms number.

When it applies. The $\frac{1}{\sqrt{2}}$ rule is for a pure sine wave. Other shapes give other ratios. A square wave that switches between plus and minus the same value has an rms value equal to its peak. Many cheap meters are calibrated to read rms only for sine waves, while true-rms meters can measure any shape.

An everyday picture

Think of a stove burner that switches on and off very quickly. What decides how hot a pan gets is the average power delivered over time, not the peak power when the burner is on. RMS is the AC version of this idea: one steady number that matches the heating the changing voltage can produce.

Worked example: Peak voltage of 120 V and 230 V mains

A US outlet is labeled $120\ \text{V}$ rms, and a European outlet is labeled $230\ \text{V}$ rms. Find the peak voltage of each, and then the peak-to-peak voltage of the US outlet.

Rearrange the formula. Start from $V_{\text{rms}} = \frac{V_{\text{peak}}}{\sqrt{2}}$ . To find the peak, multiply both sides by $\sqrt{2}$ , which undoes the division. This gives $V_{\text{peak}} = V_{\text{rms}} \times \sqrt{2}$ . Whatever we do to one side of an equation we must do to the other, so the equation stays true.
Write the value of $\sqrt{2}$ . The square root of 2 is about 1.41421, a number that, multiplied by itself, gives almost exactly 2. We use 1.41421 in the multiplications below and round the answers to one decimal place, because the voltages are given to whole volts.
Find the US peak. Multiply 120 by 1.41421: $V_{\text{peak}} = 120\ \text{V} \times 1.41421 \approx 169.7\ \text{V}$ . Multiplying by $\sqrt{2}$ makes the number about 41 percent bigger, which takes the rms value up to the highest point the wave reaches. So the US peak is about 170 V.
Find the European peak. Multiply 230 by 1.41421: $V_{\text{peak}} = 230\ \text{V} \times 1.41421 \approx 325.3\ \text{V}$ . The European peak is higher than the US peak because its rms voltage is higher, and the same square-root factor applies to both outlets.
Check the units. A voltage multiplied by a plain number with no unit is still a voltage. So both peaks are in volts, which is the correct unit for a voltage, and the square root of 2 changes only the size of the number.
Check the US case by reversing the step. Divide the peak by the square root of 2: $169.7\ \text{V} / 1.41421 \approx 120\ \text{V}$ . Getting back the labeled rms value confirms that the arithmetic is right.
Find the peak-to-peak voltage. The wave goes from its most negative point to its most positive point, so the peak-to-peak value is twice the peak: $2 \times 169.7\ \text{V} \approx 339.4\ \text{V}$ . For the US outlet that is about 339 V, which is roughly twice the peak and is a different number from the peak itself.

Answer: The US peak is about 170 V and the European peak is about 325 V. The peak-to-peak voltage of the US outlet is about 339 V. Insulation and parts must handle these peaks, not only the labeled rms number.

The numbers in this example are recomputed by script every time the site is built.

Common mistakes

Using the peak value as if it were the rms value, or the reverse. A 120 V rms supply is not a 120 V peak supply, because its peak is about 170 V.
Averaging the AC voltage over a cycle to find its effective value. The average of a sine wave is zero, so the root mean square is needed instead.
Confusing peak with peak-to-peak. The peak-to-peak voltage of a 120 V rms supply is about 340 V, which is twice the peak, so the two numbers differ by a factor of two.
Assuming a meter shows the peak voltage. Many AC meters read the rms value, so for a sine wave the peak is about 1.414 times the reading.

Where you will meet it

Mains voltage labels, where 120 V in the US and 230 V in Europe are rms values.
Choosing the voltage ratings of cables, capacitors and transformers, which must handle the peak voltage.
Power calculations for AC heaters and lamps, where the rms voltage gives the correct average power for a resistive load.
Audio and signal measurements, where an rms level describes the strength of a waveform.

Questions people ask

Why not use the average voltage?

The average of a sine wave over a full cycle is zero, because the positive and negative halves cancel. The rms value avoids that cancellation, so it measures the heating effect of the voltage.

Is the rms value the same for every waveform?

No. The $\frac{1}{\sqrt{2}}$ rule works for a pure sine wave. A square wave that switches between plus and minus the same value has an rms value equal to its peak.

Why is it called 120 V if the voltage reaches about 170 V?

Because 120 V is the rms value. The voltage reaches about 170 V at the peak of each half cycle, but its heating effect matches 120 V of direct current.

Which value is on a mains label?

The label usually gives the rms value, because that is the number that matches the power delivered. The peak is higher, and it sets the insulation and component ratings.

Electric Power: Electric power tells you how fast a device uses energy. You get it by multiplying the voltage, the push, by the current, the flow, and the answer is measured in watts.
Ohm's Law: Ohm’s law says the push on a wire, called voltage, equals the flow through it, called current, times how hard the wire resists that flow, called resistance.
Energy and Kilowatt-Hours: Energy is power multiplied by time, so a heater that uses 2 kilowatts for 3 hours uses 6 kilowatt-hours of energy.