Parallel Circuit Resistance Calculator and Formula

This calculator finds the equivalent resistance of resistors connected in parallel: enter the resistance of each branch, in ohms, and it returns their combined resistance. The result is always lower than the resistance of the smallest individual branch when all resistances are positive.

Formula

For resistors connected in parallel, the reciprocal of the equivalent resistance is the sum of the reciprocals of the branch 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 branches. Conductance is the reciprocal of resistance: G = 1/R, measured in siemens (S). The calculator accepts a comma-separated list of positive branch resistances in ohms.

Worked Example

Using the calculator’s default values of 100 Ω, 200 Ω, and 300 Ω:

1. Find each reciprocal: 1/100 = 0.01 S, 1/200 = 0.005 S, and 1/300 ≈ 0.003333 S.
2. Add the conductances: 0.01 + 0.005 + 0.003333… = 0.018333… S.
3. Take the reciprocal: R_eq = 1/0.018333… ≈ 54.55 Ω.
4. Total conductance is 1/54.545… ≈ 0.0183 S.

Why Parallel Resistance Gets Smaller

A parallel circuit gives current multiple paths between the same two connection points. Adding a branch provides another path for charge to flow, so the whole network conducts more readily. Since resistance measures opposition to current, the equivalent resistance decreases as another positive-resistance branch is added.

In a parallel connection, each branch has the same voltage across it. The source current divides among the branches according to their resistances: a lower-resistance branch carries more current, while a higher-resistance branch carries less. The branch currents add to the total current drawn from the source.

This is different from a series circuit, where the same current passes through each component and the resistances add directly. For series resistors, R_eq = R₁ + R₂ + …; do not use that rule for parallel branches.

Using the Result in Circuit Calculations

Equivalent resistance replaces a group of parallel resistors with one ideal resistor that has the same voltage-current relationship at the group’s terminals. It is useful when simplifying a circuit, estimating the current drawn from a voltage source, or checking a design calculation.

• For a voltage source, use Ohm’s law: I_total = V/R_eq, where voltage V is in volts and current I is in amperes.
• For a known total current, the voltage across the network is V = I_totalR_eq.
• To find an individual branch current, use I_i = V/R_i, using the voltage across the parallel group.

For the example, if the network is connected across an ideal 12 V source, its total current is 12/54.55 ≈ 0.22 A. The branch currents are 12/100 = 0.12 A, 12/200 = 0.06 A, and 12/300 = 0.04 A; together they total 0.22 A. Real sources and wiring may add resistance, so a physical circuit can differ from this ideal calculation.

Shortcuts, Units, and Limits

For exactly two resistors, the reciprocal formula can be rearranged to a convenient product-over-sum rule:

R_eq = (R₁R₂)/(R₁ + R₂)

For example, 100 Ω in parallel with 200 Ω gives (100 × 200)/(100 + 200) = 66.67 Ω. This shortcut applies to two resistors; for three or more, use the reciprocal-sum formula or combine pairs step by step.

Keep all resistance entries in the same unit before calculating. One kilohm (kΩ) is 1,000 Ω, and one megohm (MΩ) is 1,000,000 Ω. The result can be expressed in Ω, kΩ, or MΩ by converting after the calculation. Ohms and siemens are SI units, so there is no separate US customary resistance unit to substitute.

The formula assumes the listed components behave as fixed, ideal resistors and are genuinely connected across the same two nodes. It does not account for temperature changes, component tolerances, wiring resistance, or frequency-dependent behavior in reactive components. If a branch contains a complex network, simplify that branch first or analyze the full circuit with an appropriate circuit method.

Common Mistakes and Checks

• Adding the resistances: Direct addition is for series connections, not parallel ones.
• Forgetting the reciprocal: Add 1/R for each branch, then take the reciprocal of that sum.
• Mixing units: Convert kΩ and MΩ to a consistent unit before entering values.
• Rounding too early: Keep extra digits during intermediate steps and round the final equivalent resistance.
• Ignoring the basic check: For positive parallel resistances, the equivalent must be less than the smallest branch resistance. A larger result usually signals a formula or entry error.

A zero-ohm branch represents an ideal short circuit across the terminals; the equivalent resistance is then zero, and the reciprocal formula as written cannot accept that entry. In a real circuit, a short can cause excessive current limited by the source and other circuit elements, so it is not merely another ordinary resistor value.

Frequently Asked Questions

What is the formula for resistance in parallel?
Use 1/R_eq = 1/R₁ + 1/R₂ + …, then invert the sum to get R_eq.

Is equivalent resistance in parallel always lower?
For positive resistors connected in parallel, it is lower than the smallest individual resistance. Each added branch gives current another path.

How do I calculate two resistors in parallel?
Use R_eq = (R₁R₂)/(R₁ + R₂), with both resistances in the same unit.

Do parallel resistors have the same current?
No. They share the same voltage, but branch current depends on resistance: I = V/R. The currents in all branches add to the source current.

Can I combine resistors with different values?
Yes. Use the reciprocal formula for any number of positive resistor values, provided they are connected across the same two nodes and use consistent units.

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list r | Branch resistances | 100, 200, 300
out req | Equivalent resistance | ohm | 1 / sum(1 / r) | 2
out conductance | Total conductance | S | 1 / req | 4