Why is one minute of angle very close to 1" at 100 yards?

Question

In the shooting sports, a very common measurement is 1 minute of angle measured at 100 yards, a very common distance to shoot a rifle. If we take the tangent of 1/60 (one minute), multiply by 100 yards, and divide by 36 (to convert from yards to inches), we get:

1.047197580733"

I was just wondering if there's something "magical" about this number, in that it is very close to 1 inch, or is that mere coincidence?

Thanks, Jay

Answer 1

Note that $100$ yards is $3600$ inches and one minute is $\frac {2\pi}{60\times 360}$ radians. Estimating $2\pi =6$ this comes out at approx $\frac 1{3600}$ radian. For small $\theta$ measured in radians we have $\tan \theta \approx \theta$.

Expect error from estimate $\pi\approx 3$ to be of the order of $4.7\%$ (true angle is bigger)

Answer 2

It is a coincidence of the involved units.

More specifically, it is a coincidence of the combination of:

1. the angular unit minutes of angle, or MOA
   • which is $\frac{1}{60} \times \frac{2\pi}{360}$ $\approx 0.000291\ \mathrm{rad}$
2. the number of inches in 1 yard
   • which is $36$
   • whose inverse is $\frac{1}{36} \approx 0.0278$

You can already see the similarity between those two to decimal numbers, less a factor of $100$ (that's the $100$ yards).

For the angle $1$ MOA, its projected length $l$ at a distance of $100\ \mathrm{yd}$ is $$ \begin{align} l & = \tan{\left(\frac{1}{60} \times \frac{2\pi}{360}\right)} \times 100\ [\mathrm{yd}] \times 36\ [\mathrm{in}/\mathrm{yd}] \\ & \approx 1.047\ [\mathrm{in}] \end{align} $$

Note that the formula above is exactly the calculation you described that you did.

Whatever comes out of this expression must be a consequence of its constituents, and its constituents are the two unitless constants listed under points 1 and 2 above (and the 100 yards). Since a seemingly coincidental result, $\approx 1$ inch, does come out, such a coincidence must be a consequence of the two constants.

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...almost: Under 1. above, I gave the constant $\frac{1}{60} \times \frac{2\pi}{360}$, but the formula used the tangent of that constant. Well, the two values only differ at the 8th decimal place, since $\tan{x} \approx x$ for small values of $x$, provided that the angle is measured in radians. That's the Small-angle approximation.

Using that approximation and decimal values, we can rewrite the expression for $l$ as follows:

$$ \begin{align} l & = 0.000291\ \times 100\ [\mathrm{yd}]\ \times \frac{1}{0.0278}\ [\mathrm{in}/\mathrm{yd}] \\ & = \left( 0.0291\ \times \frac{1}{0.0278}\ \right) [\mathrm{in}] \\ & \approx 1.047\ [\mathrm{in}] \end{align} $$

From the intermediate expression we can see that the result will come out close to 1.

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For this coincidental conveience, shooters can thank the Babylonians for the MOA angular unit, the British Impreium for the choice of number of inches per yards, and God for the Small-angle approximation.