# Resistivity

> A wire's resistance grows with its length, shrinks with its thickness, and depends on what the wire is made of.

Page: https://formulabro.yunusemrevurgun.com/electricity/resistivity/  
Category: Electricity  
Published: 2026-10-22  
Updated: 2026-10-22

## Formula

$$
R = \rho \frac{L}{A}
$$

Read it out loud: "R equals rho times L, divided by A. Read R as the resistance of the wire, in ohms. Rho is the Greek letter rho, said roh, and it is the resistivity of the material. L is the length of the wire, measured along the current. A is the cross-sectional area, the area of a slice cut straight across the wire."

## Before you start

- **Powers of ten and scientific notation, like $1.68 \times 10^{-8}$**: Very small or very large numbers are written as a short number times a power of ten. The raised **exponent** says how many times to multiply by 10. A negative exponent means divide by 10 that many times. So $10^{-8}$ is 0.00000001, a decimal point followed by seven zeros and then a 1. Then $1.68 \times 10^{-8}$ is 1.68 with the decimal point moved eight places to the left, which is 0.0000000168. This form is called **scientific notation**.
- **Resistivity $\rho$ and the ohm meter ($\Omega\ \text{m}$)**: The Greek letter $\rho$, said roh, is the usual name for **resistivity**. It measures how strongly a material resists current, and it is a property of the material itself. Its unit is the **ohm meter**, written $\Omega\ \text{m}$. Multiplying it by a length in meters and dividing by an area in square meters leaves an answer in ohms, because the meters cancel.
- **Area and the square meter ($\text{m}^2$)**: Area is how much flat space a shape covers. A **square meter**, written $\text{m}^2$, is the area of a square one meter long on each side. The **cross-sectional area** of a wire is the area of a slice cut straight across it, like the round end of a pencil. A thin household wire may have a slice of only a few square millimeters.
- **Pi ($\pi$), circles and the radius**: The Greek letter $\pi$, said pie, is the number about 3.14159. It is the distance around any circle divided by the distance across it. The **radius** $r$ is the distance from the center of a circle to its edge, which is half the **diameter**, the distance straight across. The area of a circle is $\pi r^2$: pi times the radius times itself.
- **The prefix milli (m), as in millimeters**: A **millimeter**, written mm, is one thousandth of a meter, so 1 mm = 0.001 m. The prefix **milli** always means divide by 1000. Formulas that use meters need every length converted first. A wire of 1.0 mm diameter must become 0.001 m before it goes into the formula.

More notation help: https://formulabro.yunusemrevurgun.com/math-basics/

## What each symbol means

- $R$: The **resistance** of the whole wire, measured in ohms ($\Omega$). It tells how much the wire opposes the flow of current. A short piece of thick copper wire may have a resistance well under one ohm, while a long, thin wire can have many ohms.
- $\rho$: The **resistivity** of the material, measured in ohm meters ($\Omega\ \text{m}$). It belongs to the material, so every piece of the same copper has the same value. Copper at 20 $^\circ$C is about $1.68 \times 10^{-8}\ \Omega\ \text{m}$, a very small number. Rubber and other insulators have values billions of times larger, which is why they wrap wires.
- $L$: The **length** of the wire along the direction the current flows, measured in meters (m). A meter is roughly one long stride. The 10 m wire in the example is about the length of a small bus, so it is long compared with a household appliance cord.
- $A$: The **cross-sectional area**, the area of a slice cut straight across the wire, measured in square meters ($\text{m}^2$). For a round wire, $A = \pi r^2$. A thin household wire has an area of a few square millimeters, which is about $10^{-6}\ \text{m}^2$, a very small number.

## The formula piece by piece

- $\rho$: Start with the material. $\rho$ is the resistivity, a number that says how hard the material is to push current through. Copper has a small value, so copper is a good conductor, and insulators have very large values for the same job.
- $\frac{L}{A}$: The shape of the wire. $L$ is its length and $A$ is the area of its slice. Dividing the length by the area gives a number that grows when the wire is long and thin, and shrinks when it is short and thick. This ratio is the part of the wire that does not depend on the material.
- $R = \rho \frac{L}{A}$: The whole formula in one sentence: the resistance of a wire equals its material's resistivity multiplied by its length divided by its cross-sectional area. Each part of the wire enters once, with the material on the outside and the shape on the inside.

## What the formula is doing, and why

A wire's resistance depends on three things: the material it is made of, how long it is, and how thick it is. The formula $R = \rho \frac{L}{A}$ puts those three together. The answer $R$ is in ohms ($\Omega$). The Greek letter $\rho$ stands for the material, $L$ for the length, and $A$ for the thickness through the area.

**Why multiply by the length?** A longer wire is a longer path for the current, and each extra meter adds more material for the moving electrons to bump into. The resistance grows in step with the length. Double the length and you double the resistance, because the current has to push through twice as much material.

**Why divide by the area?** A thicker wire has more room for the current to pass through, like extra lanes on a highway. Double the area and the resistance halves, because the same current spreads across twice the room. This is why a thick copper cable carries large currents with little loss, while a thin wire of the same length can get hot.

**What the material does.** The resistivity $\rho$ depends on the material's atoms and on how easily electrons move through them. Copper, silver and aluminum have small values, so they conduct well. Because $\rho$ multiplies everything else, a material with twice the resistivity gives twice the resistance for a wire of the same size and length.

**When the formula does not apply.** It assumes the same material all along the wire, a current that spreads evenly across each cross section, and a fixed temperature. Resistivity changes with temperature, so table values are quoted at a stated temperature, usually 20 $^\circ$C. If the thickness of a conductor changes along its length, split it into many thin slices, find each slice's resistance and add them up.

## An everyday picture

Think of water flowing through a pipe. A long pipe makes the water work harder to get through, so the resistance grows with the length. A wide pipe lets much more water through than a narrow one at the same pressure, so the resistance falls as the area grows. A rough inside surface slows the water more than a smooth one of the same size, and that difference plays the role of the material's resistivity. The length, the width and the roughness together decide how hard the pipe is to push water through.

## Worked example: Resistance of a 10 m copper wire

A copper wire is 10 m long, roughly the length of a small bus. Its diameter is 1.0 mm, so its radius is 0.5 mm. Copper at 20 $^\circ$C has resistivity $\rho = 1.68 \times 10^{-8}\ \Omega\ \text{m}$. The wire is a practice example, like the copper inside a lamp cord. Find its resistance in ohms.

1. **Convert every length to meters.** The formula needs lengths in meters, and the wire is already 10 m long. The diameter is 1.0 mm, and a millimeter is one thousandth of a meter, so the radius, which is half the diameter, is $r = 0.5\ \text{mm} = 0.0005\ \text{m}$. In powers of ten that is $5 \times 10^{-4}$ m, because the decimal point moves four places to the left.
2. **Find the cross-sectional area.** The cut through the wire is a circle with radius $r$, and the area of a circle is $\pi r^2$, which means pi times the radius multiplied by itself. Squaring the radius gives $0.0005 \times 0.0005 = 0.00000025 = 2.5 \times 10^{-7}\ \text{m}^2$. Multiplying by pi gives $A = \pi \times 2.5 \times 10^{-7} \approx 7.85 \times 10^{-7}\ \text{m}^2$, about 0.785 square millimeters.
3. **Find the ratio of length to area.** Write 10 as $1 \times 10^{1}$ so that the powers of ten are easy to see. Dividing by a power of ten means subtracting exponents: $\frac{L}{A} = \frac{10}{7.85 \times 10^{-7}} = \frac{1}{7.85} \times 10^{1 - (-7)} = 0.1274 \times 10^{8} \approx 1.274 \times 10^{7}\ \text{m}^{-1}$. The unit $\text{m}^{-1}$ is one per meter, because meters divided by square meters leaves one over a meter.
4. **Multiply by the resistivity and check the units.** Now $R = \rho \times \frac{L}{A} = 1.68 \times 10^{-8}\ \Omega\ \text{m} \times 1.274 \times 10^{7}\ \text{m}^{-1}$. Multiply the front numbers, $1.68 \times 1.274 \approx 2.140$, and add the exponents, $-8 + 7 = -1$. The meters cancel, since $\text{m} \times \text{m}^{-1} = 1$, which leaves ohms: $R \approx 2.140 \times 10^{-1}\ \Omega = 0.214\ \Omega$.
5. **Compare with a longer wire.** Double the length to 20 m, and the resistance doubles, because the current must push through twice as much material. The result is $2 \times 0.214\ \Omega = 0.428\ \Omega$, so about 0.43 ohms. Length goes straight into the formula in step with the resistance, so the rule is easy to predict.
6. **Compare with a thicker wire.** Double the diameter to 2.0 mm, and the radius doubles too. The area depends on the radius squared, so the area becomes $2 \times 2 = 4$ times bigger. The resistance falls to a quarter, $0.214\ \Omega / 4 = 0.0535\ \Omega$. A thicker wire gives the current more room, so the same length resists much less.

**Answer:** The wire has a resistance of about **0.214 ohms** ($0.214\ \Omega$). Doubling its length would double this to about 0.43 ohms, and doubling its diameter would cut it to about 0.0535 ohms. A result in the hundreds of ohms would signal a unit mistake, such as leaving millimeters in the formula.

## Common mistakes

- Using the diameter where the radius belongs. The area is $\pi r^2$, so a diameter of 1.0 mm means a radius of 0.5 mm, half as big. Using 1.0 mm as the radius would make the area four times too large and the resistance four times too small.
- Mixing units. Resistivity uses ohm meters, so convert the length and the diameter to meters before you divide. A length in centimeters or a diameter in millimeters gives a wrong answer, often by a large factor.
- Confusing resistivity with conductivity. Conductivity is the reciprocal of resistivity, which means one divided by it, so $\sigma = 1/\rho$. A good conductor has a small resistivity and a large conductivity.
- Forgetting that resistivity depends on temperature. Table values are quoted at a stated temperature, usually 20 $^\circ$C, and a hot wire has a different value.

## Where you will meet it

- Choosing the wire thickness for house wiring, so the cable does not overheat or lose too much voltage over its length.
- Designing heating elements. Alloys with high resistivity, such as nichrome, are used because they warm up when current flows through them.
- Building strain gauges, where stretching a thin wire makes it longer and thinner, so its resistance changes in a measurable way.
- Designing thin-film and integrated resistors, where the length and width of a thin strip set its resistance.

## Questions people ask

### What is the unit of resistivity?

The ohm meter, written $\Omega\ \text{m}$. The ohm is the unit of resistance. Multiplying it by a length in meters and dividing by an area in square meters leaves ohms, because the meters cancel out.

### Why is the length on top and the area on the bottom?

A longer wire has more material for the current to push through, so the resistance goes up. A bigger area gives the current more room, so the resistance goes down. The formula puts each effect where it belongs.

### Is copper the best conductor?

Silver has a lower resistivity than copper, but it costs far more. Copper conducts very well and is cheap enough for most wiring, which is why it is so common in homes.

### Does the formula work for any shape?

It works best for a uniform wire with the same cross section along its whole length. For a shape where the thickness changes, split it into many thin slices, find each slice's resistance, and add them up.

## Related formulas

- [Ohm's Law](https://formulabro.yunusemrevurgun.com/electricity/ohms-law/)
- [Temperature Coefficient of Resistance](https://formulabro.yunusemrevurgun.com/electricity/temperature-coefficient-of-resistance/)
- [Drift Velocity](https://formulabro.yunusemrevurgun.com/electricity/drift-velocity/)
- [Resistors in Series and Parallel](https://formulabro.yunusemrevurgun.com/electricity/resistors-in-series-and-parallel/)

## Sources

- [Electrical resistivity and conductivity, Wikipedia](https://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity)
- [Electrical resistance and conductance, Wikipedia](https://en.wikipedia.org/wiki/Electrical_resistance_and_conductance)
- [Copper, Wikipedia](https://en.wikipedia.org/wiki/Copper)

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