The power triangle, also known as Watt's triangle, is a tool used in the theory of electricity to calculate electrical power in an alternating current (AC) circuit using the trigonometric properties of a right triangle.
The triangle is made up of three sides: apparent power (S), active power (P) and reactive power (Q).

Apparent power is the total power delivered to the circuit and is measured in voltamperes (VA).

Active power is the power actually used by the circuit and is measured in watts (W).

Reactive power is the power lost in the circuit due to resistance and reactance and is measured in voltamperes reactive (VAR).
The relationship between these three powers can be visualized in the triangle of powers. The apparent power (S) is the hypotenuse of the triangle and is formed from the combination of the active (P) and reactive (Q) powers, which are the other two sides of the triangle. The active power (P) forms the cosine angle with the apparent power, while the reactive power (Q) forms the sine angle with the apparent power.
What is the power triangle used for?
The power triangle is an important tool in the theory of electricity, since it allows us to understand the relationships between the different electrical powers in an alternating current (AC) circuit.
Basic applications include:

Calculation of electrical power: The power triangle allows calculating the total electrical power delivered to a circuit, as well as the active power (which is converted into useful work) and the reactive power (which does not perform useful work, but is necessary for the operation of some electrical equipment).

Circuit Performance Optimization – This triangle is useful for understanding how electrical power is distributed in a circuit and how circuit components can be optimized to improve their performance and efficiency.

Design of electrical systems: it is a useful tool for the design of electrical systems, since it allows sizing the components of the circuit and guaranteeing that they are capable of handling the electrical powers involved.

Fault Analysis – It can also be used to analyze faults in an electrical circuit to determine if there are power losses due to resistance and reactance, which can help identify and fix problems.
Resolved exercise on the triangle of powers
Statement:
There is an electrical circuit with an apparent power of 800 VA and a power factor of 0.8. Calculate the active power and reactive power of the circuit.
Solution:
First, we can calculate the active power (P) of the circuit using the formula P = S x cos(θ), where θ is the phase angle between current and voltage. In this case, since the power factor (PF) is equal to 0.8, we know that cos(θ) = PF = 0.8. Therefore, we can calculate the active power as follows:
P = 800VA x 0.8 = 640W
The active power of the circuit is 640 watts.
Then, we can calculate the reactive power (Q) of the circuit using the formula Q = S x sin(θ). We know that the phase angle (θ) can be calculated from the power factor using the following formula:
cos(θ) = PF => sin(θ) = √(1  cos²(θ))
In this case, since FP = 0.8, we can calculate sin(θ) as follows:
sin(θ) = √(1  cos²(θ)) = √(1  0.8²) = √(0.36) = 0.6
Therefore, we can calculate the reactive power as follows:
Q = 800VA x 0.6 = 480VAR
The reactive power of the circuit is 480 voltamperes reactive.
Therefore, the active power of the circuit is 640 W and the reactive power is 480 VAR.