# Circular mil

A circular mil is a unit of area, equal to the area of a circle with a diameter of one mil (one thousandth of an inch or 0.0254 mm). It is equal to π/4 square mils or approximately 5.067×10$^{−4}$ mm$^2$. It is a unit intended for referring to the area of a wire with a circular cross section. As the definition of the unit contains π, it is easy to calculate area values in circular mils when the diameter in mils is known.
The area in circular mils, A, of a circle with a diameter of d mils, is given by the formula: $\{A\}_{\mathrm {cmil} }=\{d\}_{\mathrm {mil} }^{2}.$
In Canada and the United States, the Canadian Electrical Code (CEC) and the National Electrical Code (NEC), respectively, use the circular mil to define wire sizes larger than 0000 AWG. In many NEC publications and uses, large wires may be expressed in thousands of circular mils, which is abbreviated in two different ways: kcmil or MCM. For example, one common wire size used in the NEC has a conductor diameter of 0.5 inches, or 500 mils, and thus a cross-section of $500^{2}=250{,}000$ circular mils, written as 250 kcmil or 250 MCM, which is the first size larger than 0000 AWG used within the NEC.
1000 circular mil equals approximately 0.5067 mm$^2$, so for many purposes, a ratio of 2 MCM ≈ 1 mm$^2$ can be used with negligible (1.3%) error.

## Equivalence to other units of area

As a unit of area, the circular mil can be converted to other units such as square inches or square millimetres.
1 circular mil is approximately equal to:
* 0.7854 square mils (1 square mil is about 1.273 circular mils)
* 7.854 × 10$^{−7}$ square inches (1 square inch is about 1.273 million circular mils)
* 5.067 × 10$^{−10}$ square metres
* 5.067 × 10$^{−4}$ square millimetres
* 506.7 μm$^2$
1000 circular mils = 1 MCM or 1 kcmil, and is (approximately) equal to:
* 0.5067 mm$^2$, so 2 kcmil ≈ 1 mm$^2$ (a 1.3% error)
Therefore, for practical purposes such as wire choice, 2 kcmil ≈ 1 mm$^2$ is a reasonable rule of thumb for many applications.

### Square mils

In square mils, the area of a circle with a diameter of 1 mil is:
$A=\pi r^{2}=\pi \left({\frac {d}{2}}\right)^{2}={\frac {\pi d^{2}}{4}}={\rm {{\frac {\pi \times (1~mil)^{2}}{4}}={\frac {\pi }{4}}~mil^{2}\approx 0.7854~mil^{2}.}}$
By definition, this area is also equal to 1 circular mil, so
${\rm {1~cmil={\frac {\pi }{4}}~mil^{2}.}}$
The conversion factor from square mils to circular mils is therefore 4/π cmil per square mil:
${\rm {{\frac {4}{\pi }}{\frac {cmil}{mil^{2}}}.}}$
The formula for the area of an arbitrary circle in circular mils can be derived by applying this conversion factor to the standard formula for the area of a circle (which gives its result in square mils).
${\begin{aligned}\{A\}_{{\textrm {mil}}^{2}}&=\pi r^{2}=\pi \left({\frac {d}{2}}\right)^{2}={\frac {\pi d^{2}}{4}}&&({\text{Area in square mils}})\\[2ex]\{A\}_{\textrm {cmil}}&=\{A\}_{{\textrm {mil}}^{2}}\times {\frac {4}{\pi }}&&({\text{Convert to cmil}})\\[2ex]\{A\}_{\textrm {cmil}}&={\frac {\pi d^{2}}{4}}\times {\frac {4}{\pi }}&&({\text{Substitute area in square mils with its definition}})\\[2ex]\{A\}_{\textrm {cmil}}&=d^{2}.\end{aligned}}$

### Square inches

To equate circular mils with square inches rather than square mils, the definition of a mil in inches can be substituted:
    ${\begin{aligned}{\rm {1~cmil}}&={\rm {{\frac {\pi }{4}}~mil^{2}={\frac {\pi }{4}}~(0.001~in)^{2}}}\\[2ex]&={\rm {{\frac {\pi }{4{,}000{,}000}}~in^{2}\approx 7.854\times 10^{-7}~in^{2}}}\end{aligned}}$

### Square millimetres

Likewise, since 1 inch is defined as exactly 25.4 mm, 1 mil is equal to exactly 0.0254 mm, so a similar conversion is possible from circular mils to square millimetres:
    ${\begin{aligned}{\rm {1~cmil}}&={\rm {{\frac {\pi }{4}}~mil^{2}={\frac {\pi }{4}}~(0.0254~mm)^{2}={\frac {\pi \times 0.000\,645\,16}{4}}~mm^{2}}}\\[2ex]&={\rm {1.6129\pi \times 10^{-4}~mm^{2}\approx 5.067\times 10^{-4}~mm^{2}}}\end{aligned}}$

## Example calculations

A 0000 AWG solid wire is defined to have a diameter of exactly 0.46 inches (11.68 mm). The cross-sectional area of this wire is:

### Formula 1: circular mil

Note: 1 inch = 1000 mils
    ${\begin{aligned}d&={\rm {0.46~inches=460~mil}}\\A&=d^{2}~{\rm {cmil/mil^{2}=(460~mil)^{2}~cmil/mil^{2}=211{,}600~cmil.}}\end{aligned}}$
(This is the same result as the AWG circular mil formula shown below for n = −3)

### Formula 2: square mil

    ${\begin{aligned}d&={\rm {0.46~inches=460~mils}}\\r&={d \over 2}={\rm {230~mils}}\\A&=\pi r^{2}={\rm {\pi \times (230~mil)^{2}=52{,}900\pi ~mil^{2}\approx 166{,}190.25~mil^{2}}}\end{aligned}}$

### Formula 3: square inch

    ${\begin{aligned}d&={\rm {0.46~inches}}\\r&={d \over 2}={\rm {0.23~inches}}\\A&=\pi r^{2}={\rm {\pi \times (0.23~in)^{2}=0.0529\pi \approx 0.16619~in^{2}}}\end{aligned}}$

### Calculating diameter from area

When large diameter wire sizes are specified in kcmil, such as the widely used 250 kcmil and 350 kcmil wires, the diameter of the wire can be calculated from the area without using π:
We first convert from kcmil to circular mil
    ${\begin{aligned}A&={\rm {250~kcmil=250{,}000~{\text{cmil}}}}\\d&={\sqrt {A~\mathrm {mil^{2}/cmil} }}\\d&={\rm {{\sqrt {(250{,}000~cmil)~mil^{2}/cmil}}=500~mil=0.500~inches}}\end{aligned}}$
Thus, this wire would have a diameter of a half inch or 12.7 mm.

## Metric equivalent

Some tables give conversions to circular millimetres (cmm). The area in cmm is defined as the square of the wire diameter in mm. However, this unit is rarely used in practice. One of the few examples is in a patent for a bariatric weight loss device.
    ${\rm {1~cmm=\left({\frac {1000}{25.4}}\right)^{2}~cmil\approx 1{,}550~cmil}}$

## AWG circular mil formula

The formula to calculate the area in circular mil for any given AWG (American Wire Gauge) size is as follows. $A_{n}$ represents the area of number $n$ AWG.
    $A_{n}=\left(5\times 92^{(36-n)/39}\right)^{2}$
For example, a number 12 gauge wire would use $n=12$:
    $\left(5\times 92^{(36-12)/39}\right)^{2}=6530~{\textrm {cmil}}$
Sizes with multiple zeros are successively larger than 0 AWG and can be denoted using "number of zeros/0"; for example "4/0" for 0000 AWG. For an $m$/0 AWG wire, use
    $n=-(m-1)=1-m$ in the above formula.
For example, 0000 AWG (4/0 AWG), would use $n=-3$; and the calculated result would be 211,600 circular mils.

## Standard sizes

Standard sizes are from 250 to 400 in increments of 50 kcmil, 400 to 1000 in increments of 100 kcmil, and from 1000 to 2000 in increments of 250 kcmil.
The diameter in the table below is that of a solid rod with the given conductor area in circular mils. Stranded wire is larger in diameter to allow for gaps between the strands, depending on the number and size of strands.
Standard kcmil wire sizes & solid copper equivalents :
| Area | Area | Diameter | Diameter | NEC copper wire ampacity with 60/75/90 °C insulation (A) | NEC copper wire ampacity with 60/75/90 °C insulation (A) | NEC copper wire ampacity with 60/75/90 °C insulation (A) |
| (kcmil, MCM) | (mm$^2$) | (in) | (mm) | NEC copper wire ampacity with 60/75/90 °C insulation (A) | NEC copper wire ampacity with 60/75/90 °C insulation (A) | NEC copper wire ampacity with 60/75/90 °C insulation (A) |
| 250 | 126.7 | 0.500 | 12.70 | 215 | 255 | 290 |
| 300 | 152.0 | 0.548 | 13.91 | 240 | 285 | 320 |
| 350 | 177.3 | 0.592 | 15.03 | 260 | 310 | 350 |
| 400 | 202.7 | 0.632 | 16.06 | 280 | 335 | 380 |
| 500 | 253.4 | 0.707 | 17.96 | 320 | 380 | 430 |
| 600 | 304.0 | 0.775 | 19.67 | 355 | 420 | 475 |
| 700 | 354.7 | 0.837 | 21.25 | 385 | 460 | 520 |
| 750 | 380.0 | 0.866 | 22.00 | 400 | 475 | 535 |
| 800 | 405.4 | 0.894 | 22.72 | 410 | 490 | 555 |
| 900 | 456.0 | 0.949 | 24.10 | 435 | 520 | 585 |
| 1000 | 506.7 | 1.000 | 25.40 | 455 | 545 | 615 |
| 1250 | 633.4 | 1.118 | 28.40 | 495 | 590 | 665 |
| 1500 | 760.1 | 1.225 | 31.11 | 520 | 625 | 705 |
| 1750 | 886.7 | 1.323 | 33.60 | 545 | 650 | 735 |
| 2000 | 1013.4 | 1.414 | 35.92 | 560 | 665 | 750 |
Note: For smaller wires, consult American wire gauge § Tables of AWG wire sizes.