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When you plug a toaster into a standard 120-volt wall outlet, the voltage actually surges up to nearly +170 volts and plunges down to -170 volts sixty times every second.
We label the outlet "120 V" because alternating current delivering fluctuating voltage heats the toaster coil at the exact same rate as a steady 120-volt direct current source.
How do we prove mathematically that this effective voltage is always the peak voltage divided by the square root of two?

The Three Steps of RMS

The term RMS stands for root-mean-square, which tells you the mathematical operations to perform in reverse order: square the function, take the mean (average) over one period, and take the square root.
Let's work through the calculus to see where the 2\sqrt{2} factor comes from.

Integrating Instantaneous Power

Consider a sinusoidal voltage v(t)=V0sin⁑(Ο‰t)v(t) = V_0 \sin(\omega t), where V0V_0 is the peak voltage and Ο‰=2Ο€T\omega = \frac{2\pi}{T} is the angular frequency for a cycle of period TT.
By definition, the mean of the squared voltage over one full cycle is given by the definite integral Vrms2=1T∫0Tv(t)2 dt=V02T∫0Tsin⁑2(Ο‰t) dtV_{\text{rms}}^2 = \frac{1}{T} \int_0^T v(t)^2 \, dt = \frac{V_0^2}{T} \int_0^T \sin^2(\omega t) \, dt
To integrate sin⁑2(Ο‰t)\sin^2(\omega t), we use the trigonometric double-angle identity sin⁑2(ΞΈ)=1βˆ’cos⁑(2ΞΈ)2\sin^2(\theta) = \frac{1 - \cos(2\theta)}{2}, which converts the squared sine term into a simple cosine oscillation.