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). It corresponds to 5.067×10⁻⁴ mm². It is a unit intended for referring to the area of a wire with a circular cross section. As the area in circular mils can be calculated without reference to π, the unit makes conversion between cross section and diameter of a wire considerably easier.

The area in circular mils, A, of a circle with a diameter of d mils, is given by the formula:

$A= d^2$

Electricians in Canada and the United States are familiar with the circular mil because the National Electrical Code (NEC) uses 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: MCM^[1] or kcmil.^[2] For example, one common wire size used in the NEC has a cross-section of 250,000 circular mils, written as 250 kcmil or 250 MCM, which is the first size larger than 0000 AWG used within the NEC.

Equivalence to other units of area

Although square mils are rarely used, it is convenient to convert between square inches and circular mils. As a classic example taken from the NEC, a 0000 AWG solid wire is defined to have a diameter of exactly 0.46 inch.

Formula 1: Circular Mil

Note: 1 inch = 1000 mils

$d= 0.46$ inch = 460 mils

$A= d^2$

$A= 460^2$ = 211,600 circular mils

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

Formula 2: Square Mil

$d= 0.46$ inch = 460 mils

$r= {d \over 2}$ = 230 mils

$A= \pi r^2$

$A= \pi \times 230^2 = 52,900 \pi \approx 166,190.25$ square mils

Formula 3: Square Inch

$d= 0.46$ inch

$r= {d \over 2}$ = 0.23 inch

$A= \pi r^2$

$A= \pi \times (0.23)^2 = .0529 \pi \approx 0.16619$ square inches

Formula 4: Solving for Circular Mil

$A= A$

$211,600$ circular mils $= {846,400 \over \pi}$ square mils

1 circular mil $= {846,400 \over 211,600\pi}$ square mils

1 circular mil $= {4 \over \pi}$ square mils

Formula 5: Solving for Square Mil

1 square mil $= {\pi \over 4}$ circular mils

Therefore, the following conversions apply:

• To obtain square mils   ⇒ (# of circular mils) × π ÷ 4
• To obtain square inches ⇒ (# of circular mils) × π ÷ 4,000,000
• To obtain circular mils ⇒ (# of square mils)   × 4 ÷ π
• To obtain circular mils ⇒ (# of square inches) × 4,000,000 ÷ π

1 circular mil is approximately equal to:

• 1.273 square mils (1 square mil is about 0.7854 circular mils)
• 7.854×10⁻⁷ square inches (1 square inch is about 1.273 million circular mils)
• 5.067×10⁻¹⁰ square metres
• 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.

Note: the conversion to metric actually has an infinite number of digits. The following shows 55 digits:

1_circular_mil= pi * (1e-3 /2 * 25.4)**2 = 506.7074790974977514316397512891510201921614524252108178e-6 mm2

1_MCM = 1000 circular_mil = 506.7074790974977514316397512891510201921614524252108178e-3 mm2

When large diameter wire sizes are specified in kcmil, such as the ubiquitous 250 kcmil or 350 kcmil wires, the wire's diameter can be easily determined with the following formula:

Formula 6: Diameter

Note: We first convert from kcmil to circular mil

$A= 250$ kcmil

$A= 250,000$ circular mils

$d= \sqrt{A}$

$d= \sqrt{250,000}$ = 500 mils = 0.500 inch

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

AWG circular mil formula

The formula to calculate the circular mil for any given AWG (American Wire Gauge) size is as follows. A_n represents the circular mil area for the AWG size n.

$A_n =(5 \times 92^\frac{36-n}{39})^2$

• For example, a number 12 gauge wire would use n = 12; and the calculated result would be 6529.946789 circular mils

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

n = −(m−1) = 1−m in the above formula.

For example, the number 0000 gauge or 4/0 gauge, 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.^[3]

The diameter in the table below is that of a solid rod with the given conductor area in circular mils. Stranded wire is around 5% 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

kcmil MCM	mm²	Diameter		NEC copper wire ampacity with 60/75/90 °C insulation (A)[4]
		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 the Table of AWG wire sizes article.

Circular inch

Cardarelli has asserted that the "circular inch" (cin) was used as a unit for the measurement of wire sizes, with a conversion of 1 circular inch = 1,000,000 circular mil.^[5] The Oxford English Dictionary does not acknowledge the use of "circular inch", although it has an entry for "circular mil".^[6]

See also

• thou (length)
• square mil

References

1. "Power Plant Siting Acronyms", California Energy Commission
2. "Popular Acronyms". NEMA
3. NFPA 70-2011 National Electrical Code 2011 Edition. 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).
4. NFPA 70 National Electrical Code 2008 Edition. 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 NFPA and the original complete Code must be consulted. In particular, the maximum permissible overcurrent protection devices may set a lower limit.
5. Cardarelli, François Cradarelli (2003). Encyclopaedia of Scientific Units, Weights and Measures. Springer. p. 32. ISBN 978-1-4471-1122-1.
6. Full Oxford English Dictionary consulted online 1 March 2015, under "Circular, adj. Special uses: S1 Technical"