The Power Triangle: kW, kVA, and kVAR Explained
Calculate the AC power triangle: enter real power (kW) and reactive power (kVAR) to get apparent power (kVA) and power factor instantly.
kVA = √(kW² + kVAR²), PF = kW / kVA
Common conversions
| Input | Result |
|---|---|
| 5 kW / 3 kVAR | 5.83 kVA, PF 0.86 |
| 8 kW / 6 kVAR | 10.00 kVA, PF 0.80 |
| 10 kW / 7.5 kVAR | 12.50 kVA, PF 0.80 |
| 12 kW / 5 kVAR | 13.00 kVA, PF 0.92 |
| 15 kW / 20 kVAR | 25.00 kVA, PF 0.60 |
| 20 kW / 15 kVAR | 25.00 kVA, PF 0.80 |
| 24 kW / 7 kVAR | 25.00 kVA, PF 0.96 |
| 50 kW / 20 kVAR | 53.85 kVA, PF 0.93 |
The math behind it
- kVA = √(60² + 45²)
- kVA = √(3600 + 2025) = √5625
- kVA = 75
- PF = 60 / 75 = 0.80
- φ = arccos(0.80) ≈ 36.87°
Everything you need to know
The power triangle is a right triangle that shows how three kinds of AC power relate to each other. It turns an abstract idea, real work versus wasted circulation, into a shape you can measure with a ruler and a calculator. Every AC load, from a light bulb to a factory's whole switchgear, can be plotted as one triangle.
What each side of the triangle represents
The adjacent side is real power, measured in kW. This is the power that actually does something: it lights a bulb, spins a motor shaft, or heats a coil. The opposite side is reactive power, measured in kVAR. Inductors and capacitors pull this power from the source and hand it back every AC cycle without ever converting it into light, heat, or motion. The hypotenuse is apparent power, measured in kVA, the total current-carrying capacity the source has to supply so the load gets both the real power it uses and the reactive power it borrows and returns. Because the three sides form a right triangle, they follow the Pythagorean relationship: kVA² = kW² + kVAR².
The phase angle and power factor
The angle between the kW side and the kVA hypotenuse is called φ (phi), the phase angle. It measures how far the current waveform has shifted, in time, from the voltage waveform. When φ is 0°, current and voltage rise and fall together, all of the apparent power is real power, and the triangle collapses flat with no kVAR leg at all. As φ grows, more of the hypotenuse tilts away from the kW axis into kVAR territory. The cosine of that angle is the power factor: PF = cos(φ) = kW / kVA. A PF of 1.0 means φ = 0°. A PF of 0.80, common for a loaded induction motor, corresponds to φ ≈ 36.87°.
Why reactive power still matters
Reactive power is not wasted energy in the sense of heat lost to friction. It is energy stored in a magnetic or electric field for a fraction of a cycle and then returned to the source. But the current that carries it still flows through every wire, breaker, and transformer winding between the source and the load. A cable sized only for real power will run hot and a generator rated only in kW will stall out on kVA capacity long before its kW rating is reached, because the reactive current takes up room the real current needs.
Shrinking the triangle with power factor correction
Capacitor banks installed near an inductive load supply local reactive power instead of pulling it from the utility feeder. That shortens the kVAR leg of the triangle without changing the kW leg, which pulls the phase angle toward zero and the power factor toward 1.0. The practical result is a smaller kVA hypotenuse for the same useful output, freeing up transformer and conductor capacity and, on many commercial rate schedules, avoiding a low-power-factor penalty charge.
Common applications
Facility managers plot their monthly kW and kVAR from utility meter data to find average power factor and estimate whether a low-PF surcharge applies, then size correction capacitors from the shortfall.
Standby generators and UPS units are rated in kVA, not kW. Knowing the load's real kW and reactive kVAR lets you compute the true kVA demand instead of assuming PF 1.0 and undersizing the unit.
Correcting a motor or plant from a known starting PF to a target PF (often 0.95) requires calculating the kVAR difference between the two triangles, which sets the capacitor bank's required kVAR rating.
Engineers use the triangle to check whether a feeder's apparent-power draw is approaching the transformer's kVA nameplate, even when the connected kW load looks well within range.
Common mistakes
kW and kVAR are perpendicular quantities and can't be added arithmetically. Combining them requires the Pythagorean relationship, kVA = √(kW² + kVAR²), not a simple sum.
Only purely resistive loads like heaters have PF = 1. Motors, transformers, and electronic drivers all draw reactive current, so treating every load as PF 1.0 understates the true kVA demand.
Inductive loads (most motors) lag, meaning current trails voltage; capacitive loads lead. Mixing up the sign when combining multiple loads on a feeder can produce a badly wrong total kVAR estimate.
A generator rated 100 kVA at 0.8 PF only delivers 80 kW of real output. Reading the kVA number as if it were the usable kW capacity is a common and costly sizing error.
Frequently asked questions
What is the power triangle in electrical engineering?+
The power triangle is a right-triangle diagram showing how real power (kW), reactive power (kVAR), and apparent power (kVA) relate. Real power is the horizontal leg, reactive power is the vertical leg, and apparent power is the hypotenuse, linked by kVA² = kW² + kVAR².
Yes or no: is kVA always greater than or equal to kW?+
Yes, apparent power (kVA) is always equal to or greater than real power (kW). They're only equal when reactive power is zero and the power factor is exactly 1.0, which happens only with a purely resistive load.
How do you find kVA from kW and kVAR?+
Take the square root of the sum of the squares: kVA = √(kW² + kVAR²). For example, 30 kW combined with 40 kVAR gives kVA = √(900 + 1600) = √2500 = 50 kVA.
What is the phase angle φ in the power triangle?+
Phi (φ) is the angle between the real-power leg and the apparent-power hypotenuse, equal to the time-shift between the voltage and current waveforms. Its cosine equals the power factor, so PF = cos(φ). A φ of 25.8° corresponds to a power factor of about 0.90.
No, does reactive power do any useful work?+
No, reactive power (kVAR) does not perform work like turning a shaft or producing light. It is energy that inductors and capacitors store and release each AC cycle, but it still generates current that loads down wiring and transformers.
Can reactive power be negative?+
Yes, by convention inductive loads (motors, transformers) draw positive kVAR and are called lagging, while capacitive loads supply negative kVAR and are called leading. A capacitor bank effectively subtracts kVAR from an inductive load's total.
What is a typical power factor for an induction motor?+
A fully loaded induction motor typically runs at a power factor between 0.80 and 0.90. At light load, the same motor's power factor can drop to 0.35-0.50, because the magnetizing current stays nearly constant while the real-power draw falls.
How do you reduce the kVAR leg of the power triangle?+
Install capacitor banks close to the inductive load to supply the reactive current locally instead of drawing it from the utility feeder. Correcting a plant from PF 0.75 to PF 0.95 can shrink the required kVA by roughly 20% for the same kW output.
Why do utilities charge penalties for a low power factor?+
A low power factor forces the utility to size generation, transformers, and feeders for a higher kVA than the customer's actual kW usage requires. Many commercial rate schedules add a surcharge once average power factor drops below about 0.90-0.95.
How is the power triangle different from an impedance triangle?+
The power triangle plots kW, kVAR, and kVA and its angle is the same φ as the impedance triangle, but the impedance triangle plots resistance (R), reactance (X), and impedance (Z) in ohms instead of power in watts. The two triangles are similar because power is proportional to the square of current times impedance.