Three-phase power runs factories, large buildings, and heavy motors. Its formulas look like single-phase power with one extra factor, the square root of 3, and one extra confusion: line values versus phase values. Get those two right and the rest is arithmetic.
The core formulas
Real power P = square root of 3 x line voltage x line current x power factor. Apparent power S = square root of 3 x line voltage x line current. Reactive power Q = square root of 3 x line voltage x line current x sin(power factor angle). Power factor = P / S. Units: P in watts, S in volt-amps, Q in volt-amps reactive. Divide by 1,000 for kW, kVA, kVAR.
The square root of 3 (about 1.732) appears because three phases share the load. Forgetting it is the classic error: using single-phase P = V x I on a three-phase system understates power by 42%.
Line versus phase: Wye and Delta
Line voltage is measured between two phases; phase voltage is measured between one phase and neutral. In a Wye (star) system, line voltage = square root of 3 x phase voltage, and line current = phase current. In a Delta system, line voltage = phase voltage, and line current = square root of 3 x phase current.
Example: a 400-volt Wye system has a phase voltage of 400 / 1.732 = 231 volts. That is why European systems quote 400 volts line and 230 volts phase for the same supply. Mixing line and phase values in one formula is the second classic error.
Worked examples
Balanced load, 400 volts line, 10 amps line, power factor 1.0: P = 1.732 x 400 x 10 x 1.0 = 6,928 watts, about 6.93 kW. Apparent power is the same, 6.93 kVA, since the power factor is 1.
Motor example, 480 volts line, 50 amps line, power factor 0.85: S = 1.732 x 480 x 50 = 41,568 volt-amps, about 41.6 kVA. P = 41.6 x 0.85 = 35.4 kW. Reactive power Q = square root of (S squared – P squared) = about 21.9 kVAR. Wye phase voltage = 480 / 1.732 = 277 volts.
Solving backwards: current from power
To find line current from a known power: I = P / (square root of 3 x line voltage x power factor). Example: a 50 kW motor at 480 volts, power factor 0.90. I = 50,000 / (1.732 x 480 x 0.90) = 50,000 / 748.2 = 66.8 amps. Size cables, breakers, and contactors from this current with the usual safety margins and local rules, never from the power figure alone.
Balanced versus unbalanced loads
All the standard formulas assume a balanced load: equal current in all three phases. Real installations are often unbalanced, with one phase carrying more load than the others. In that case the three-phase formulas give only an approximation.
For unbalanced loads, calculate each phase separately with single-phase maths: phase power = phase voltage x phase current x phase power factor, then add the three phase powers. The neutral conductor then carries the imbalance current, which is why neutral sizing matters in unbalanced systems. If your phases differ by more than about 10%, rebalance the loads across phases before trusting any three-phase total.
Try it faster with the tool
You can follow every step above by hand, but the free Three-Phase Power Calculator does the arithmetic for you. Enter your numbers and it gives the answer right away, so you can check your hand calculation or skip straight to the result.
Quick tips
- P = root 3 x line volts x line amps x power factor.
- Wye: line volts = root 3 x phase volts. Delta: line volts = phase volts.
- Never mix line and phase values in one formula.