It's all about power. We want 100 volts AC and 100 volts DC to heat a resistor to the same temperature. A worthy goal. The RMS formula gives us that "corrected" AC voltage. Average voltage and peak voltage are useless in terms of POWER. We are about to find out why.
Our analysis need only consider a quarter of a cycle. The other three-quarters yield the same area under the curve, the same squares, the same power.
Power is a function of the square of the voltage (Power=V²/R). Finding the square of a DC voltage is easy. How do we find the square of a voltage changing as a sine wave. You chop up the sine wave and square the voltage at each slice. Graphically, you can take each voltage slice and draw it as a square shooting out into the z-axis. Add up all the square areas, then divide by the number of squares. That gives you the average square area (mean of the squares) for that sine wave with a peak Vp. Now we take the square root of thethat average square area to get the voltage that produces it. And that is, in simplified graphical terms, the Vrms formula. We found the average sized square for the power formula and from that found the length of the edge of the square (Vrms).
