RMS Voltage Calculator
A sine wave’s voltage is a moving target — so the world quotes its RMS value, the DC equivalent that delivers the same power. Enter a peak voltage, and this calculator returns the RMS, the peak-to-peak swing and the rectified average.
The DC Equivalent of a Moving Target
RMS — root mean square — answers the only question about AC that pays bills: what steady DC voltage would deliver the same average power to a resistor? Square the wave (making both halves positive and weighting the peaks), average over a cycle, and take the root: for a sine, Vrms = Vpeak ÷ √2. Every AC rating on earth is this number in disguise — the 230 V of a socket is RMS, the amplifier's honest watts are computed from RMS, and insulation, by contrast, must survive the peak, which is why the calculator prints both: the same wave, two different engineering questions (heating versus stress). The third figure, the rectified average (2Vp ÷ π), is what cheap average-responding meters actually sense before pretending, via a 1.11 form-factor fudge, that they measured RMS — a polite fiction that holds only for the sine this page assumes.
Worked Example: The 325 Volt Illusion
The defaults enter the European mains' peak, 325 V: RMS = 325 ÷ 1.41421 = 229.8 V — the 230 on every nameplate — while peak-to-peak shows the wave's full 650 V swing and the rectified average reads 206.9 V (the raw sense of an uncorrected average meter). North America's 120 V RMS is a 170 V peak by the same division. The worked example's lesson is the grid's double bookkeeping: a heater fed from the socket consumes as if 230 steady volts drove it, while a capacitor or a semiconductor across the same socket must be rated past 325 — and a designer who confuses the two figures builds either a cold room or a small fire. The calculator exists to keep the two ledgers one click apart. A caution for the bench: oscilloscopes display peak (and peak-to-peak) by nature, multimeters display RMS by convention, and the same bench wave therefore wears two numbers differing by exactly this page's √2 — the commonest source of "the supply is wrong" panics in electronics, resolved by remembering which instrument speaks which dialect, a bilingualism this page exists to teach.
Why the Form Factor Betrays Non-Sines
The ratio Vrms ÷ Vavg(rect) — 1.11 for a sine — is shape's fingerprint: square waves give 1.00, triangles 1.15, and the spiky current waveforms of modern electronics (switch-mode supplies sipping current at the waveform's crests) can reach 2 or beyond, which is where average-responding meters under-report dangerously and true-RMS instruments earn their price. Crest factor (peak ÷ RMS) is the companion spec the oscilloscope age cares about. This page computes the sine's canonical trio and names the neighbouring shapes' ratios in its note, because a calculator that converts peak to RMS owes its reader the warning that the conversion is a property of the waveform, not of electricity in general.
The Heating Element's Indifference
A resistive heater is the RMS concept's perfect citizen: it cannot tell 230 V RMS AC from 230 V DC, delivering identical heat to either, because its physics is v² averaged — the definition itself. Edison's DC lamps and Westinghouse's AC lamps met on exactly this ground when ratings were standardised: brightness per rated volt had to match across systems, and RMS was the treaty number. Modern echoes abound — an inverter's "modified sine" output (a stepped square-ish wave) heats a heater honestly at its stated RMS while confusing motors and meters that trusted the sine ratios; the calculator's form-factor warnings are, in that appliance drawer, live practical advice rather than pedantry.
The Square Wave's Revenge in the Workshop
Feed a cheap inverter's stepped output — RMS-honest at 230 V but peaking nearer 270 than 325 — into equipment designed around the sine's ratios and curious faults bloom: power supplies whose DC rails run low (they charge toward the peak, not the RMS), motors humming hotter (their iron sees a harmonic banquet the sine's form factor never budgeted), timers and dimmers misbehaving by the waveform's edges. The calculator's clean sine arithmetic is the reference these faults are diagnosed against: measure the actual peak and RMS of the offending wave, compare with this page's ratio of 1.414, and the deviation is the diagnosis. Standards for "true sine" inverters exist because the ratio is a compatibility contract, and this page states the contract's terms.
Related Tools
More from the Physics Calculators hub: the Wave Velocity Calculator — the wave whose voltage this is — and the live Voltage Drop Calculator, for what the wire does to this RMS on its way to the load.
Frequently Asked Questions
Is 230 V the peak or the RMS of my mains?
RMS, always, in every country's nameplate convention: the peak of a 230 V supply is about 325 V (this page's worked example), of a 120 V supply about 170 V. Equipment input stages see the peak (a rectified supply charges its capacitor to near it), heaters and motors see the RMS (power's measure), and the fuse sees the RMS current. When a datasheet says a part is "rated 250 VAC," the rating is an RMS working-voltage figure with the peak handled inside its test regime — one more reason the two numbers must never be casually swapped.
What does a multimeter actually display on AC?
On a true-RMS meter, the computed root-mean-square (via a thermal sensor or digital sampling) — accurate for any waveform within its bandwidth and crest-factor specs. On an average-responding meter (most inexpensive ones), the rectified average multiplied by 1.11 — the sine's form factor — so it displays "RMS" correctly only for undistorted sines and under-reads the distorted currents of LED drivers, variable-speed drives and PC supplies, sometimes by 30% or more. The calculator's three figures let you reproduce either instrument's reading from the peak and see the discrepancy coming.
Why square, average, then root — why not a plain average?
Because power is proportional to v²: the plain average of a sine over a cycle is zero (the halves cancel), which answers no useful question, while the mean of the squares is exactly the average power per ohm. The order of operations is the physics: heating does not care about sign, so the wave is squared first; the root at the end restores volts from volts-squared so the answer can stand in Ohm's-law arithmetic as an effective DC value. RMS is less a statistic than a bookkeeping treaty between AC and DC, and every term of it is load-bearing.
What about RMS current and power?
The identical construction applies (Irms = Ipeak ÷ √2 for a sine), and P = Vrms × Irms holds for resistive loads — the average power, the billing quantity. With reactive loads the product of RMS figures gives apparent power (VA), and the real power wears the cos φ discount of the power-factor literature. The calculator covers voltage as titled; the symmetry is stated here because readers habitually need the current twin, and its arithmetic could not be simpler than "the same ratios."
Does the peak matter for audio amplifiers?
The peak is the amplifier's ceiling story: the supply rails fix the maximum peak the output can swing (a ±40 V rail stage clips near 40 V peak), from which the honest sine RMS output into a load follows by this page's division — 28.3 V RMS — and the watts by V²rms ÷ R. Marketing watts are often computed at the peak ("peak power" = twice the RMS power for a sine) or at clipping's square-wave-ish extremes; the calculator's arithmetic, applied to any brochure, separates the physics from the poetry in one division by √2.
What is the RMS of a pulse or a DC-offset wave?
The general definition never changes — square, mean, root — but the closed forms do: a unipolar pulse train of peak V and duty cycle D has Vrms = V√D, and a sine riding on a DC offset combines the two in quadrature (Vrms = √(Vdc² + Vac,rms²)), the Pythagorean bookkeeping of power adding across components. These are the cases where this page's sine-only ratios must be set aside and the defining integral honoured — a digital oscilloscope's RMS measurement does exactly that numerically. The calculator's jurisdiction is the pure sine stated in its labels; this FAQ prices the common departures so the reader recognises when the √2 shortcut has left its territory.
Why do datasheets quote both Vrms and Vpeak for capacitors and semiconductors?
Because the two numbers guard different failures: RMS current heats the device's resistive paths (the ageing, thermal failure), while peak voltage stresses its insulation and junctions (the puncture, breakdown failure) regardless of how briefly it lasts. A part can be well inside its RMS comfort and still be murdered by the peak — the worked example's 325 V hiding inside a "230 V" supply being the canonical trap — and the calculator's habit of printing both side by side is the defence this page teaches by layout.
Who standardised RMS?
The quantity grew with the AC industry in the 1890s — the war of the currents was fought in RMS ratings, since DC's constant volts needed an AC equivalent to compete on lamp brightness and motor torque — and the form-factor tables of the early electrical handbooks canonised the sine's 0.7071. Steinmetz's complex-number machinery made the whole AC calculus routine, RMS included. The calculator's ratios are those handbooks' frontispiece numbers, computed fresh from the peak each run rather than looked up — the modern courtesy.