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How to Calculate Minute of Angle

The minute of angle denotes an amount of circular arc subtended by the radius of the circle upon which the arc lies. It is also known as the minute of arc (MOA) or the arcminute. A single minute of angle is the equivalent of 1/60th of one angular degree. This small value makes the MOA an ideal unit for use in applications that deal with very long distances in which such small angular deviations are significant. Included in these applications is the accuracy of firearms, cartography, property surveying and astronomy measurements. A calculation of the minute of angle follows from simple (but exaggerated) trigonometry. A test of firearms accuracy will illustrate the process, given a target that has already been hit.

Things You'll Need

Trigonometry-capable calculator

Instructions

- 1
  Draw a diagram of the firearms test design. This should look like a triangle where the hypotenuse connects the bullet origin (the firearm barrel) to the spot actually hit on the target. The longer non-hypotenuse side should extend from the bullet's origin to the center of the target. The shorter non-hypotenuse side will connect the the center of the target to the point actually hit on the target. The angle to be measured is the one between the hypotenuse and the longer non-hypotenuse segment of the triangle.
- 2
  Label the lengths of the two non-hypotenuse sides of the triangle. The longer will just be the distance from the bullet's origin to the target. The smaller will be the distance measured between the center of the target and the struck point on the target.
- 3
  Write down the equation x = (y)tan(MOA/60). This is the relationship among the minute of angle and the distances involved in the calculation. In this equation, x is equal to the distance of the shorter side of the triangle while y is equal to the distance of the longer non-hypotenuse side of the triangle.
- 4
  Solve the equation you wrote down in Step 3 for the MOA in the following way: Divide both sides by y, then take the inverse tangent (arctan) of both sides, and finally multiply both sides by 60. From this you will get a new equation: MOA = 60arctan(x/y).
- 5
  Substitute your distances (for x and y--given in Step 3) into the right-hand side of the equation derived in Step 4. Run this equation through a trig-capable calculator set to degree mode. The proper execution of this step should give you about 1 minute of arc (rounded up) for x = 1 inch and y = 3,600 inches.

Astronomy