Circular mil#Metric equivalent

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{{sup|2}}

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

• 0.5067 mm{{sup|2}}, so 2 kcmil ≈ 1 mm{{sup|2}} (a 1.3% error)

Therefore, for practical purposes such as wire choice, 2 kcmil ≈ 1 mm{{sup|2}} is a reasonable rule of thumb for many applications.

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

$$\backslash rm\; 1~cmil\; =\; \backslash frac\{\backslash pi\}\{4\}~mil^2.$$

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

$$\backslash rm\; \backslash frac\{4\}\{\backslash pi\}\; \backslash frac\{cmil\}\{mil^2\}.$$

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{align}

\{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{align}

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

:$$

\begin{align}

\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{align}

Likewise, since 1 inch is defined as exactly 25.4{{nbsp}}mm, 1{{nbsp}}mil is equal to exactly 0.0254{{nbsp}}mm, so a similar conversion is possible from circular mils to square millimetres:

:$$

\begin{align}

\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{align}

A 0000 AWG solid wire is defined to have a diameter of exactly {{convert|0.46|in|mm|sigfig=4}}. The cross-sectional area of this wire is:

Note: 1 inch = 1000 mils

:$\backslash begin\{align\}$

d &= \rm 0.46~inches = 460~mil \\

A &= d^2 ~ \rm cmil/mil^2 = (460~mil)^2 ~ cmil/mil^2 = 211{,}600~cmil.

\end{align}

(This is the same result as the AWG circular mil formula shown below for {{math|1=n = −3}})

:$\backslash begin\{align\}$

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{align}

:$\backslash begin\{align\}$

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{align}

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 {{pi}}:

We first convert from kcmil to circular mil

:$\backslash begin\{align\}$

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{align}

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

Some tables give conversions to circular millimetres (cmm).Charles Hoare, The A.B.C. of Slide Rule Practice, p. 52, London: Aston & Mander, 1872 {{oclc|605063273}}Edwin James Houston, A Dictionary of Electrical Words, Terms and Phrases, p. 135, New York: W. J. Johnston, 1889 {{oclc|1069614872}} 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.Greg A. Lloyd, Bariatric Magnetic Apparatus and Method of Manufacturing Thereof, US patent {{patent|US|8481076}}, 9 July 2013.

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

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.

:$A\_n\; =\; \backslash left\; (5\; \backslash times\; 92^\{(36\; -\; n)/39\}\backslash right)^2$

For example, a number 12 gauge wire would use $n\; =\; 12$:

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

Sizes with multiple zeros are successively larger than 0{{nbsp}}AWG and can be denoted using "number of zeros/0"; for example "4/0" for 0000{{nbsp}}AWG. For an $m$/0{{nbsp}}AWG wire, use

:$n\; =\; -(m\; -\; 1)\; =\; 1\; -\; m$ in the above formula.

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

Standard sizes are from 250 to 400 in increments of 50{{nbsp}}kcmil, 400 to 1000 in increments of 100{{nbsp}}kcmil, and from 1000 to 2000 in increments of 250{{nbsp}}kcmil.[http://bulk.resource.org/codes.gov/ NFPA 70-2011 National Electrical Code 2011 Edition] {{Webarchive|url=https://web.archive.org/web/20081015074713/http://bulk.resource.org/codes.gov/ |date=2008-10-15 }}. Table 310.15(B)(17) page 70-155, Allowable Ampacities of Single-Insulated Conductors Rated Up to and Including 2000 Volts in Free Air, Based on Ambient Air Temperature of 30°C (86°F).

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 {{slink|American wire gauge|Tables of AWG wire 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

{{reflist|2}}

Category:Units of area