Convert second of arc to angular mil

Learn how to convert 1 second of arc to angular mil step by step.

Calculation Breakdown

Set up the equation
\(1.0\left(second \text{ } of \text{ } arc\right)={\color{rgb(20,165,174)} x}\left(angular \text{ } mil\right)\)
Define the base values of the selected units in relation to the SI unit \(\left(radian\right)\)
\(\text{Left side: 1.0 } \left(second \text{ } of \text{ } arc\right) = {\color{rgb(89,182,91)} \dfrac{π}{6.48 \times 10^{5}}\left(radian\right)} = {\color{rgb(89,182,91)} \dfrac{π}{6.48 \times 10^{5}}\left(rad\right)}\)
\(\text{Right side: 1.0 } \left(angular \text{ } mil\right) = {\color{rgb(125,164,120)} 2.0 \times \dfrac{π}{6.4 \times 10^{3}}\left(radian\right)} = {\color{rgb(125,164,120)} 2.0 \times \dfrac{π}{6.4 \times 10^{3}}\left(rad\right)}\)
Insert known values into the conversion equation to determine \({\color{rgb(20,165,174)} x}\)
\(1.0\left(second \text{ } of \text{ } arc\right)={\color{rgb(20,165,174)} x}\left(angular \text{ } mil\right)\)
\(\text{Insert known values } =>\)
\(1.0 \times {\color{rgb(89,182,91)} \dfrac{π}{6.48 \times 10^{5}}} \times {\color{rgb(89,182,91)} \left(radian\right)} = {\color{rgb(20,165,174)} x} \times {\color{rgb(125,164,120)} {\color{rgb(125,164,120)} 2.0 \times \dfrac{π}{6.4 \times 10^{3}}}} \times {\color{rgb(125,164,120)} \left(radian\right)}\)
\(1.0 \cdot {\color{rgb(89,182,91)} \dfrac{π}{6.48 \times 10^{5}}} \cdot {\color{rgb(89,182,91)} \left(rad\right)} = {\color{rgb(20,165,174)} x} \cdot {\color{rgb(125,164,120)} 2.0 \times \dfrac{π}{6.4 \times 10^{3}}} \cdot {\color{rgb(125,164,120)} \left(rad\right)}\)
\(\text{Cancel SI units}\)
\(1.0 \times {\color{rgb(89,182,91)} \dfrac{π}{6.48 \times 10^{5}}} \cdot {\color{rgb(89,182,91)} \cancel{\left(rad\right)}} = {\color{rgb(20,165,174)} x} \times {\color{rgb(125,164,120)} 2.0 \times \dfrac{π}{6.4 \times 10^{3}}} \times {\color{rgb(125,164,120)} \cancel{\left(rad\right)}}\)
\(\text{Conversion Equation}\)
\(\dfrac{π}{6.48 \times 10^{5}} = {\color{rgb(20,165,174)} x} \times 2.0 \times \dfrac{π}{6.4 \times 10^{3}}\)
Cancel factors on both sides
\(\text{Cancel factors}\)
\(\dfrac{{\color{rgb(255,204,153)} \cancel{π}}}{6.48 \times {\color{rgb(99,194,222)} \cancelto{10^{2}}{10^{5}}}} = {\color{rgb(20,165,174)} x} \times \dfrac{{\color{rgb(255,204,153)} \cancel{π}} \times 2.0}{6.4 \times {\color{rgb(99,194,222)} \cancel{10^{3}}}}\)
\(\dfrac{1.0}{6.48 \times 10^{2}} = {\color{rgb(20,165,174)} x} \times \dfrac{2.0}{6.4}\)
Switch sides
\({\color{rgb(20,165,174)} x} \times \dfrac{2.0}{6.4} = \dfrac{1.0}{6.48 \times 10^{2}}\)
Isolate \({\color{rgb(20,165,174)} x}\)
Multiply both sides by \(\left(\dfrac{6.4}{2.0}\right)\)
\({\color{rgb(20,165,174)} x} \times \dfrac{2.0}{6.4} \times \dfrac{6.4}{2.0} = \dfrac{1.0}{6.48 \times 10^{2}} \times \dfrac{6.4}{2.0}\)
\({\color{rgb(20,165,174)} x} \times \dfrac{{\color{rgb(255,204,153)} \cancel{2.0}} \times {\color{rgb(99,194,222)} \cancel{6.4}}}{{\color{rgb(99,194,222)} \cancel{6.4}} \times {\color{rgb(255,204,153)} \cancel{2.0}}} = \dfrac{1.0 \times 6.4}{6.48 \times 10^{2} \times 2.0}\)
\({\color{rgb(20,165,174)} x} = \dfrac{6.4}{6.48 \times 10^{2} \times 2.0}\)
Rewrite equation
\(\dfrac{1.0}{10^{2}}\text{ can be rewritten to }10^{-2}\)
\({\color{rgb(20,165,174)} x} = \dfrac{10^{-2} \times 6.4}{6.48 \times 2.0}\)
Solve \({\color{rgb(20,165,174)} x}\)
\({\color{rgb(20,165,174)} x}\approx0.0049382716\approx4.9383 \times 10^{-3}\)
\(\text{Conversion Equation}\)
\(1.0\left(second \text{ } of \text{ } arc\right)\approx{\color{rgb(20,165,174)} 4.9383 \times 10^{-3}}\left(angular \text{ } mil\right)\)

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