Circular mil

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 μm²

1000 circular mils = 1 MCM or 1 kcmil, and is (approximately) equal to:

• 0.5067 mm², so 2 kcmil ≈ 1 mm² (a 1.3% error)

Therefore, for practical purposes such as wire choice, 2 kcmil ≈ 1 mm² is a reasonable rule of thumb for many applications.

In square mils, the area of a circle with a diameter of 1 mil is:

$${\displaystyle A=\pi r^{2}=\pi \left({\frac {d}{2}}\right)^{2}={\frac {\pi d^{2}}{4}}={\rm {{\frac {\pi \times (1~mil)^{2}}{4}}={\frac {\pi }{4}}~mil^{2}\approx 0.7854~mil^{2}}}}$$

By definition, this area is also equal to 1 circular mil, so

$${\displaystyle {\rm {1~cmil={\frac {\pi }{4}}~mil^{2}}}}$$

The conversion factor from square mils to circular mils is therefore 4/π cmil per square mil:

$${\displaystyle {\rm {1~mil^{2}={\frac {4}{\pi }}~cmil}}}$$

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).

$${\displaystyle {\begin{aligned}A_{{\textrm {mil}}^{2}}&=\pi r^{2}=\pi \left({\frac {d}{2}}\right)^{2}={\frac {\pi d^{2}}{4}}&&({\text{Area in square mils}})\\[2ex]A_{\textrm {cmil}}&=A_{{\textrm {mil}}^{2}}\times {\frac {4}{\pi }}&&({\text{Convert to cmil}})\\[2ex]A_{\textrm {cmil}}&={\frac {\pi d^{2}}{4}}\times {\frac {4}{\pi }}&&({\text{Substitute area in square mils with its definition}})\\[2ex]A_{\textrm {cmil}}&=d^{2}&&({\text{where d is measured in mils}})\\[2ex]\end{aligned}}}$$

To equate circular mils with square inches rather than square mils, the definition of a mil in inches can be substituted:

${\displaystyle {\begin{aligned}{\rm {1~cmil}}&={\rm {{\frac {\pi }{4}}~mil^{2}={\frac {\pi }{4}}~(0.001~in)^{2}}}\\[2ex]&={\rm {{\frac {\pi }{4{,}000{,}000}}~in^{2}\approx 7.854\times 10^{-7}~in^{2}}}\end{aligned}}}$

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:

${\displaystyle {\begin{aligned}{\rm {1~cmil}}&={\rm {{\frac {\pi }{4}}~mil^{2}={\frac {\pi }{4}}~(0.0254~mm)^{2}={\frac {\pi \times 0.000\,645\,16}{4}}~mm^{2}}}\\[2ex]&={\rm {1.6129\pi \times 10^{-4}~mm^{2}\approx 5.067\times 10^{-4}~mm^{2}}}\end{aligned}}}$

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:

Note: 1 inch = 1000 mils

${\displaystyle {\begin{aligned}d&={\rm {0.46~inches=460~mil}}\\A&=d^{2}~{\rm {cmil/mil^{2}=(460~mil)^{2}~cmil/mil^{2}=211{,}600~cmil.}}\end{aligned}}}$

(This is the same result as the AWG circular mil formula shown below for n = −3)

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

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

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

${\displaystyle {\begin{aligned}A&={\rm {250~kcmil=250{,}000~{\text{cmil}}}}\\d&={\sqrt {A~\mathrm {mil^{2}/cmil} }}\\d&={\rm {{\sqrt {(250{,}000~cmil)~mil^{2}/cmil}}=500~mil=0.500~inches}}\end{aligned}}}$

Thus, this wire would have a diameter of a half inch or 12.7 mm.

Some tables give conversions to circular millimetres (cmm).^[3][4] 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.^[5]

${\displaystyle {\rm {1~cmm=\left({\frac {1000}{25.4}}\right)^{2}~cmil\approx 1{,}550~cmil}}}$

The formula to calculate the area in circular mil for any given AWG (American Wire Gauge) size is as follows. ${\displaystyle A_{n}}$ represents the area of number ${\displaystyle n}$ AWG.

${\displaystyle A_{n}=\left(5\times 92^{(36-n)/39}\right)^{2}}$

For example, a number 12 gauge wire would use ${\displaystyle n=12}$:

${\displaystyle \left(5\times 92^{(36-12)/39}\right)^{2}=6530~{\textrm {cmil}}}$

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 ${\displaystyle m}$/0 AWG wire, use

${\displaystyle n=-(m-1)=1-m}$ in the above formula.

For example, 0000 AWG (4/0 AWG), would use ${\displaystyle n=-3}$; and the calculated result would be 211,600 circular mils.

In North America wires larger than the AWG are available in sizes beginning with a half-inch (500 mil) diameter. However, solid core wire of that size would be quite stiff for most uses as it resists bending and coiling for transport. Therefore, most large wires are made of tightly-bound strands of smaller wire with the same cross-sectional area of conductors. The table below has a diameter column that is for solid wire with no strands. Since standard sizes have a fixed area, a stranded wire would always have a larger diameter than the table shown below.

Large standard wires range from 250 to 400 kcmil in increments of 50 kcmil, from 400 to 1000 in increments of 100 kcmil, and from 1000 to 2000 in increments of 250 kcmil.^[6]

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.

Note: For smaller wires, consult American wire gauge § Tables of AWG wire sizes. Note: Aluminum wires have a much lower ampacity than copper but are often available in these sizes.

• Thou (length)
• Square mil
• IEC 60228, the metric wire-size standard used in most parts of the world.
• American Wire Gauge (AWG), used primarily in the US and Canada
• Standard Wire Gauge (SWG), the British imperial standard BS3737, superseded by the metric.
• Stubs Iron Wire Gauge
• Jewelry wire gauge
• Body jewelry sizes
• Electrical wiring
• Number 8 wire, a term used in the New Zealand vernacular