Calculating Resistors in Parallel: Formula and Examples

This calculator finds the equivalent resistance of resistors connected in parallel: enter each resistance in ohms and get the combined resistance. For the defaults, three 10 Ω resistors in parallel have an equivalent resistance of 3.33 Ω.

Formula

For resistors connected across the same two nodes, the reciprocal of the equivalent resistance equals the sum of the reciprocals of the individual resistances:

1 / R_eq = 1 / R₁ + 1 / R₂ + … + 1 / R_n

Equivalently, R_eq = 1 / Σ(1 / R_i). Here, R_eq is equivalent resistance in ohms (Ω), R_i is the resistance of branch i in ohms, and n is the number of resistors. The calculator applies this reciprocal-sum formula to the values in the list.

For two resistors only, the same rule can be written as R_eq = (R₁ × R₂) / (R₁ + R₂). This shortcut is convenient, but the reciprocal-sum form works for any number of parallel resistors.

Worked Example

Using the calculator’s default values, R₁ = 10 Ω, R₂ = 10 Ω, and R₃ = 10 Ω:

1. Find each reciprocal: 1/10 = 0.1 Ω⁻¹.
2. Add the reciprocals: 0.1 + 0.1 + 0.1 = 0.3 Ω⁻¹.
3. Take the reciprocal of the sum: R_eq = 1/0.3 = 3.333… Ω.
4. Round to two decimal places: R_eq = 3.33 Ω.

Why Parallel Resistance Gets Smaller

Parallel components share both connection points, so they have the same voltage across them. Current has more than one path to follow; adding a branch increases the circuit’s total conductance, which is the reciprocal of resistance. Consequently, equivalent resistance decreases as another positive-resistance branch is added.

A useful check is that a parallel combination of ordinary, positive resistors must have a resistance lower than the smallest individual resistance. Three 10 Ω resistors therefore cannot combine to make 10 Ω or more. Identical resistors have an especially simple result: n equal resistors of resistance R in parallel give R/n.

How to Use the Result

Use equivalent resistance when replacing a parallel resistor network with one resistor for circuit analysis. This simplification preserves the voltage-current relationship seen at the network’s two terminals; it does not mean the original branches have disappeared physically or that each branch carries the same current.

With a known voltage V across the parallel network, total current is I_total = V/R_eq, using volts, ohms, and amperes. The current in an individual branch is I_i = V/R_i. Thus, lower-resistance branches carry more current. The equivalent resistance alone does not tell you whether a particular resistor can safely handle its power; for each branch, calculate P_i = V²/R_i and compare it with the component’s rated power, allowing appropriate design margin.

Resistance is expressed in ohms in both metric and US customary electrical practice; there is no separate customary resistance unit to convert. If a value is in kilohms, convert it to ohms by multiplying by 1,000, or keep every input in kilohms and interpret the result in kilohms. Mixing ohms and kilohms without conversion gives a misleading answer.

Common Mistakes and Limits

• Using the series rule: adding resistances directly calculates a series combination, not a parallel one.
• Misidentifying the connection: components are parallel only when each connects across the same two nodes. A drawing’s left-to-right layout does not by itself establish that.
• Rounding too early: keep full precision during reciprocal calculations, then round the final result to a useful number of significant figures.
• Entering zero: an ideal zero-ohm branch makes the equivalent resistance zero. Its reciprocal is undefined, so do not enter zero into this calculator; analyze an ideal short circuit separately. Real wires and components have nonzero parasitic resistance.
• Ignoring real-world tolerances: the formula gives a nominal result from the values entered. Actual resistance varies with component tolerance, temperature, and operating conditions.

The formula assumes resistors behave as ohmic components at the conditions considered and that the listed values are resistances, not impedances. For AC circuits with capacitors or inductors, phase and frequency matter; a simple sum of resistance reciprocals is not generally enough.

Frequently Asked Questions

What is the formula for resistors in parallel?
Take the reciprocal of each resistance, add those reciprocals, and take the reciprocal of the sum: R_eq = 1/(1/R₁ + 1/R₂ + …). Use the same resistance unit for every value.

Is the equivalent resistance always lower than the smallest resistor?
For two or more positive, finite resistors in parallel, yes. Each added branch provides another path for current and increases total conductance. An ideal zero-ohm branch is a special short-circuit case.

How do I calculate two resistors in parallel quickly?
Use R_eq = (R₁ × R₂)/(R₁ + R₂). For example, 6 Ω and 3 Ω give (6 × 3)/(6 + 3) = 2 Ω.

Do identical resistors in parallel divide resistance evenly?
The equivalent resistance is the individual resistance divided by the number of resistors. For example, four 20 Ω resistors in parallel give 5 Ω. This is a rule for resistance, not a claim that voltage is divided among parallel branches; each branch has the same voltage.

Do parallel resistors have the same current?
No. They have the same voltage across them, but branch current depends on resistance: I = V/R. Equal resistors carry equal current; unequal resistors generally do not.

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list r | Resistor values | 10, 10, 10
out req | Equivalent resistance | Ω | 1 / sum(1 / r) | 2