# Convert full turn to second of arc

Learn how to convert 1 full turn to second of arc step by step.

## Calculation Breakdown

Set up the equation
$$1.0\left(full \text{ } turn\right)={\color{rgb(20,165,174)} x}\left(second \text{ } of \text{ } arc\right)$$
Define the base values of the selected units in relation to the SI unit $$\left(radian\right)$$
$$\text{Left side: 1.0 } \left(full \text{ } turn\right) = {\color{rgb(89,182,91)} 2.0 \times π\left(radian\right)} = {\color{rgb(89,182,91)} 2.0 \times π\left(rad\right)}$$
$$\text{Right side: 1.0 } \left(second \text{ } of \text{ } arc\right) = {\color{rgb(125,164,120)} \dfrac{π}{6.48 \times 10^{5}}\left(radian\right)} = {\color{rgb(125,164,120)} \dfrac{π}{6.48 \times 10^{5}}\left(rad\right)}$$
Insert known values into the conversion equation to determine $${\color{rgb(20,165,174)} x}$$
$$1.0\left(full \text{ } turn\right)={\color{rgb(20,165,174)} x}\left(second \text{ } of \text{ } arc\right)$$
$$\text{Insert known values } =>$$
$$1.0 \times {\color{rgb(89,182,91)} 2.0 \times π} \times {\color{rgb(89,182,91)} \left(radian\right)} = {\color{rgb(20,165,174)} x} \times {\color{rgb(125,164,120)} {\color{rgb(125,164,120)} \dfrac{π}{6.48 \times 10^{5}}}} \times {\color{rgb(125,164,120)} \left(radian\right)}$$
$$\text{Or}$$
$$1.0 \cdot {\color{rgb(89,182,91)} 2.0 \times π} \cdot {\color{rgb(89,182,91)} \left(rad\right)} = {\color{rgb(20,165,174)} x} \cdot {\color{rgb(125,164,120)} \dfrac{π}{6.48 \times 10^{5}}} \cdot {\color{rgb(125,164,120)} \left(rad\right)}$$
$$\text{Cancel SI units}$$
$$1.0 \times {\color{rgb(89,182,91)} 2.0 \times π} \cdot {\color{rgb(89,182,91)} \cancel{\left(rad\right)}} = {\color{rgb(20,165,174)} x} \times {\color{rgb(125,164,120)} \dfrac{π}{6.48 \times 10^{5}}} \times {\color{rgb(125,164,120)} \cancel{\left(rad\right)}}$$
$$\text{Conversion Equation}$$
$$2.0 \times π = {\color{rgb(20,165,174)} x} \times \dfrac{π}{6.48 \times 10^{5}}$$
Cancel factors on both sides
$$\text{Cancel factors}$$
$${\color{rgb(255,204,153)} \cancel{π}} \times 2.0 = {\color{rgb(20,165,174)} x} \times \dfrac{{\color{rgb(255,204,153)} \cancel{π}}}{6.48 \times 10^{5}}$$
$$\text{Simplify}$$
$$2.0 = {\color{rgb(20,165,174)} x} \times \dfrac{1.0}{6.48 \times 10^{5}}$$
Switch sides
$${\color{rgb(20,165,174)} x} \times \dfrac{1.0}{6.48 \times 10^{5}} = 2.0$$
Isolate $${\color{rgb(20,165,174)} x}$$
Multiply both sides by $$\left(\dfrac{6.48 \times 10^{5}}{1.0}\right)$$
$${\color{rgb(20,165,174)} x} \times \dfrac{1.0}{6.48 \times 10^{5}} \times \dfrac{6.48 \times 10^{5}}{1.0} = 2.0 \times \dfrac{6.48 \times 10^{5}}{1.0}$$
$$\text{Cancel}$$
$${\color{rgb(20,165,174)} x} \times \dfrac{{\color{rgb(255,204,153)} \cancel{1.0}} \times {\color{rgb(99,194,222)} \cancel{6.48}} \times {\color{rgb(166,218,227)} \cancel{10^{5}}}}{{\color{rgb(99,194,222)} \cancel{6.48}} \times {\color{rgb(166,218,227)} \cancel{10^{5}}} \times {\color{rgb(255,204,153)} \cancel{1.0}}} = 2.0 \times \dfrac{6.48 \times 10^{5}}{1.0}$$
$$\text{Simplify}$$
$${\color{rgb(20,165,174)} x} = 2.0 \times 6.48 \times 10^{5}$$
Solve $${\color{rgb(20,165,174)} x}$$
$${\color{rgb(20,165,174)} x} = 1296000 = 1.296 \times 10^{6}$$
$$\text{Conversion Equation}$$
$$1.0\left(full \text{ } turn\right) = {\color{rgb(20,165,174)} 1.296 \times 10^{6}}\left(second \text{ } of \text{ } arc\right)$$