Resistor Calculator

A conductor's resistance follows from the material and the geometry:

$R = \\frac{\\rho L}{A} \\qquad A = \\frac{\\pi d^2}{4}$

Symbols and SI units:

• $R$ — resistance, ohms (Ω)
• $\\rho$ — resistivity, a material property, ohm-metres (Ω·m)
• $L$ — conductor length, metres (m)
• $A$ — cross-sectional area, square metres (m²)
• $d$ — conductor diameter, metres (m)

At 20 °C, copper is $1.68 \\times 10^{-8}$ Ω·m and aluminium $2.65 \\times 10^{-8}$ Ω·m.

The operating assumption: a uniform cross-section at a stated temperature, carrying DC or low-frequency AC. At high frequency the skin effect pushes current to the surface and the effective resistance rises.

The mistake people make: quoting a cable size in mm² and then a length in metres but a diameter in millimetres. Convert everything to metres first, or work consistently in Ω·mm²/m.

Resistors in series carry one current and their resistances add; resistors in parallel share one voltage and their conductances add:

$R_s = R_1 + R_2 + \\cdots \\qquad \\frac{1}{R_p} = \\frac{1}{R_1} + \\frac{1}{R_2} + \\cdots$

For exactly two in parallel the product-over-sum shortcut applies:

$R_p = \\frac{R_1 R_2}{R_1 + R_2}$

A parallel combination is always smaller than the smallest branch — a useful sanity check.

The power a resistor must get rid of is

$P = I^2 R = \\frac{V^2}{R} \\quad [\\text{W}]$

Common practice is to run a resistor at no more than about half its rated power so it stays cool. Treat that as orientation only: the binding limit is the manufacturer's derating curve for the actual ambient temperature and mounting.

The mistake people make: inverting only once. $1/R_p$ is the sum of reciprocals — you still have to invert the total at the end.

• Forgetting the final reciprocal — $1/R_p = 0.0136$ means $R_p = 73.5$ Ω, not $0.0136$ Ω.
• Mixing prefixes — $4.7$ kΩ is $4700$ Ω and $1.2$ MΩ is $1.2 \\times 10^{6}$ Ω.
• Confusing resistance with resistivity — resistivity is a property of the material in Ω·m; resistance is a property of the particular piece of wire.
• Using the radius in $A = \\pi d^2/4$ — that formula wants the diameter; with the radius use $A = \\pi r^2$.
• Ignoring temperature — copper rises about $0.393\\%$ per °C, so a conductor at $75$ °C has roughly $22\\%$ more resistance than at $20$ °C.
• Sizing a package from average power — a pulsed load can exceed the rating instantaneously even when the average looks safe.
• Reading the tolerance band as a digit — the gold or silver band is tolerance, not a value.