Small-angle formula

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The small-angle formula is a mathematical approximation, used in astronomy. It relates the linear size (D) of an object to its angular size (X) and its distance from the observer (d) and is a valid approximation if X is small.

SMALL-ANGLE APPROXIMATION. Tan(x) is approximately equal to x if x is small.

The approximate formula is:

D = X · d / 206,265

Here, X has to be measured in arcseconds.

The number 206,265 is approximately equal to the number of arcseconds in a circle (1,296,000), divided by 2π.

The exact formula is

D = 2 d tan(X·π/1,296,000)

and the above approximation follows from this because tan(X) is approximately equal to X if X is small.

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