View source for Circular mil

{{short description|Unit of area}}
{{Infobox unit
| name = circular mil
| image = circmil.jpg
| caption = 
| standard = 
| quantity = Area
| symbol = cmil
| symbol2 = 
| symbol3 = 
| namedafter = 
| extralabel = Derivation
| extradata = 1 cmil = {{sfrac|{{pi}}|4}}(0.001 in)<sup>2</sup>
| units1 = SI units
| inunits1 = {{val|506.7075|u=μm2}}
| units2 = FPS units
| inunits2 = {{val|7.853982|e=-7|u=in2}}
}}
A '''circular mil''' is a [[Units of measurement|unit]] of [[area]], equal to the area of a [[circle]] with a [[diameter]] of one [[Thou (unit of length)|mil]] (one thousandth of an inch or {{convert|1/1000|in|mm|disp=out|sigfig=3}}). It is equal to [[pi|{{pi}}]]/4 square mils or approximately {{val|5.067|e=-4|u=mm2}}. 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 {{pi}}, it is easy to calculate area values in circular mils when the diameter in mils is known.

The area in circular mils, {{mvar|A}}, of a circle with a diameter of {{mvar|d}}&nbsp;mils, is given by the formula:
<math display=block>\{A\}_\mathrm{cmil} = \{d\}_\mathrm{mil}^2.</math><!-- This is the correct formula for area IN UNITS OF CIRCULAR MILS. The formula πr² with r in mils gives the area in *square mils*, not circular mils. If you calculate the πr² area, then multiply by 4/π (the unit conversion ratio from square mils to circular mils) you get the area in circular mils. If you combine the formula for calculating area with the unit conversion into 1 step, the area A in circular mils is simply d². That is the entire point of circular mils, because the unit definition contains π, in order to cancel out with the π in the area formula. This is already explained in the article. Read it. -->

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&nbsp;[[American wire gauge|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'''<ref>[http://www.nema.org/stds/Popular-Acronyms.cfm "Popular Acronyms"] {{Webarchive|url=https://web.archive.org/web/20110903050334/http://www.nema.org/stds/Popular-Acronyms.cfm |date=2011-09-03 }}. NEMA</ref> or '''MCM'''.<ref>[https://www.energy.ca.gov/resources/energy-acronyms "Energy Acronyms"], California Energy Commission</ref> 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 <math>500^2 = 250{,}000</math> circular mils, written as 250&nbsp;kcmil or 250&nbsp;MCM, which is the first size larger than 0000 AWG used within the NEC.

1000 circular mil equals approximately {{val|0.5067|u=mm2}}, so for many purposes, a ratio of 2&nbsp;MCM ≈ 1&nbsp;mm<sup>2</sup> 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<sup>−7</sup> square inches (1 square inch is about 1.273 million circular mils)
* 5.067 × 10<sup>−10</sup> [[square metre]]s
* 5.067 × 10<sup>−4</sup> [[square millimetre]]s
* 506.7 μm{{sup|2}}

'''1000 circular mils''' = 1&nbsp;MCM or 1&nbsp;kcmil, and is (approximately) equal to:
* 0.5067&nbsp;mm{{sup|2}}, so 2&nbsp;kcmil ≈ 1&nbsp;mm{{sup|2}} (a 1.3% error)

Therefore, for practical purposes such as wire choice, 2&nbsp;kcmil ≈ 1&nbsp;mm{{sup|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:

<math display=block>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.
</math>

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

<math display=block>\rm 1~cmil = \frac{\pi}{4}~mil^2.</math>

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

<math display=block>\rm \frac{4}{\pi} \frac{cmil}{mil^2}.</math>

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).

<math display=block>
\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}
</math>

===Square inches===

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

:<math>
\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}
</math>

===Square millimetres===

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:

:<math>
\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}
</math>

==Example calculations==

A 0000&nbsp;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:

=== Formula 1: circular mil ===

'''Note: 1 inch = 1000 mils'''
:<math>\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}</math>

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

=== Formula 2: square mil ===
:<math>\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}</math>

=== Formula 3: square inch ===
:<math>\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}</math>

=== Calculating diameter from area ===

When large diameter wire sizes are specified in kcmil, such as the widely used 250&nbsp;kcmil and 350&nbsp;kcmil wires, the diameter of the wire can be calculated from the area without using {{pi}}:

We first convert from kcmil to circular mil
:<math>\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}</math>

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

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

:<math> \rm 1~cmm = \left( \frac{1000}{25.4} \right) ^2~cmil \approx 1{,}550~cmil </math>

==AWG circular mil formula==
The formula to calculate the area in circular mil for any given AWG ([[American Wire Gauge]]) size is as follows. <math>A_n</math> represents the area of number <math>n</math> AWG.

:<math>A_n = \left (5 \times 92^{(36 - n)/39}\right)^2</math>

For example, a number 12 gauge wire would use <math>n = 12</math>:  
:<math>\left(5 \times 92^{(36-12)/39}\right)^2 = 6530~\textrm{cmil}</math>

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 <math>m</math>/0{{nbsp}}AWG wire, use 
:<math>n = -(m - 1) = 1 - m</math> in the [[#Calculating diameter from area|above formula]].

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

==Standard sizes==
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.<ref>''[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)''.</ref>

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.<!--Assuming circular strands, and based on circular packing efficiency of pi/sqrt(12) = 0.9069, so sqrt(1/0.9069) = 1.050075)-->

{|class=wikitable style="text-align:center"
|+ Standard kcmil wire sizes<br/>& solid copper equivalents
|-
! colspan=2 | Area
! colspan=2 | Diameter
! rowspan=2 colspan=3 | [[National Electrical Code|NEC]] copper wire<br/>[[ampacity]] with<br/>60/75/90&nbsp;°C<br/>insulation (A)<ref>''[http://bulk.resource.org/codes.gov/ NFPA 70 National Electrical Code 2008 Edition] {{Webarchive|url=https://web.archive.org/web/20081015074713/http://bulk.resource.org/codes.gov/ |date=2008-10-15 }}''.  Table 310.16 page 70-148, ''Allowable ampacities of insulated conductors rated 0 through 2000 volts, 60°C through 90°C, not more than three current-carrying conductors in raceway, cable, or earth (directly buried) based on ambient temperature of 30°C''.  Extracts from [[NFPA 70]] do not represent the full position of [[National Fire Protection Association|NFPA]] and the original complete Code must be consulted. In particular, the maximum permissible [[overcurrent protection]] devices may set a lower limit.</ref>
|-
! (kcmil, <br/>MCM)
! (mm<sup>2</sup>)
! (in)
! (mm)
|-
|  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 {{slink|American wire gauge|Tables of AWG wire sizes}}.''

==See also==
* [[Thou (length)]]
* [[Square mil]]
* 
* [[IEC 60228]], the metric wire-size standard used in most parts of the world.
* [[American wire gauge|American Wire Gauge (AWG)]], used primarily in the US and Canada
* [[Standard wire gauge|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

==References==
{{reflist|2}}

[[Category:Units of area]]