# How to convert 10 radian in degree minutes and seconds ?

May 16, 2018

${572}^{\circ} 57 ' 36 ' '$

#### Explanation:

1. converting to degree : ( Multiply by $\frac{180}{\pi}$ )
$10 \times \frac{180}{\pi} = 572. {\textcolor{red}{96}}^{\circ}$

2. converting to minutes :( $\textcolor{red}{0.96} \times 60$ )
$\textcolor{w h i t e}{.}$
$0.96 \times 60 = 57. \textcolor{b l u e}{6}$

3. converting into seconds : ( $\textcolor{b l u e}{0.6} \times 60$ )
$\textcolor{w h i t e}{.}$
$0.6 \times 60 = 36$

So , $10$ radians $= {572}^{\circ} 57 ' 36 ' '$

May 16, 2018

10 rad $\approx {572}^{\circ} 57 ' 28$

#### Explanation:

We need to know what a radian is to answer this. An angle with an arc that has the same length as the radius of a circle is called a radian. As the circumference of a circle with radius = 1 har a length of $2 \pi$, it follows that π rad = 180°

Therefore 10 rad = $\frac{{180}^{\circ} \cdot 10}{\pi} = {572.95779513}^{\circ}$

As there are $60 '$ in ${1}^{\circ}$ and $60$” in $1 '$, we get
${0.95779513}^{\circ} = 0.95779513 \cdot 3600$$\approx 3448$$= 57 ' 28$

Therefore we get:
10 rad $\approx {572}^{\circ} 57 ' 28$

Please note that this is more than one time around the circle, by the way.

And by the way, please note that the discrepancy between my and @Evan's answer is due to the level of accuracy used.

May 16, 2018

${572}^{o} \textcolor{w h i t e}{\text{..")57^("'") color(white)("..")28^("''}}$ to the nearest second

#### Explanation:

$\textcolor{red}{\text{Important fact: there are "2pi" radians in a circle}}$

So $\underline{\text{1 radian}}$ in degrees is $\frac{360}{2 \pi} = \frac{180}{\pi} \text{ degrees}$

We have 10 radians so the count of degrees is

$10 \times \frac{180}{\pi} = \frac{1800}{\pi} \text{ Degrees} \approx 572.957795 \ldots .$

Store this in your calculator memory to reduce rounding errors.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
So the degrees part of the measure is: ${572}^{o}$

The minutes part is $\left(\frac{1800}{\pi} - 572\right) \times 60 = 57.467 . . = {57}^{'}$

The second's part is:
$\left(\frac{1800}{\pi} - 572 - \frac{57}{60}\right) = 7.79513 \ldots \times {10}^{-} 3 \text{ degrees}$

Convert this into seconds: $7.79513 \ldots \times {10}^{-} 3 \div \frac{1}{{60}^{2}} = 28.0624 \ldots . \text{seconds}$

So the answer is: ${572}^{o} \textcolor{w h i t e}{\text{..")57^("'") color(white)("..")28^("''}}$ to the nearest second