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If I want to convert from degrees to radians, I can use the function that takes degree value as an input, multiplies it with $\frac{\pi}{180}$ and returns the radian value: $\operatorname{DtoR}(d)=d \times \frac{\pi}{180}$.

And if I want to go from radians to degrees I need to only go backwards and divide radian value with $\frac{\pi}{180}$ (e.g. multiply it with $\frac{180}{\pi}$): $\operatorname{RtoD}(r)=r \times \frac{180}{\pi}$.

My question is this: Why does multiplying/dividing with $\frac{\pi}{180}$ converts degrees into radians/radians into degrees? Why exactly that number, not some other? Also, does this work only for unit circle, or for any circle?

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  • $\begingroup$ It is not clear what you are asking. The circumference of the unit circle is $2\pi$ which corresponds to 360°. $\endgroup$
    – copper.hat
    Feb 14, 2018 at 17:25
  • $\begingroup$ Either no one commented on this post, or he simply reposted it because he wasn't satisfied from the answers he initially received. $\endgroup$ Feb 14, 2018 at 17:28
  • $\begingroup$ @copper.hat Yes, for unit circle (unit circle's radius is 1). If radius of a circle is not 1, the circumference is not $2\pi$. $\endgroup$
    – Hanlon
    Feb 14, 2018 at 17:29
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    $\begingroup$ Because $180^\circ = \pi \text{ radians}. \qquad$ $\endgroup$ Feb 14, 2018 at 17:29
  • $\begingroup$ Go here $\longrightarrow$ khanacademy.org/math/algebra2/trig-functions/… $\endgroup$
    – Mr Pie
    Feb 14, 2018 at 17:30

5 Answers 5

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Take a look at these ratios:
$\frac{180}{d}=\frac{\pi}{r}$

Where $d$ is the degrees and $r$ is radians. Knowing that $\pi$ radians is 180 degrees one can setup this ratio to find the values they're looking for.

Finding degrees: $d=\frac{180}{\pi}r$
Finding radians: $r=\frac{\pi}{180}d$

These equations are simply derived from the first ratio.

It doesn't only need to be $\frac{\pi}{180}$, it can also be setup as:
$\frac{360}{d}=\frac{2\pi}{r}$
Because it is also known that $2\pi$ radians is a full revolution about the circle just as 360 degrees is.

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  • $\begingroup$ What if $2\pi \neq 360^\circ$? Check out my comment on the gt6989b's answer. $\endgroup$
    – Hanlon
    Feb 14, 2018 at 18:12
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    $\begingroup$ This never occurs? Unless the unit definitions were changed (ie. a full revolution equals 378deg), this never happens. $\pi$ is a constant, it doesn't matter how the circle is changed. As long as the circle is a circle, $\pi$ still holds true. $\endgroup$
    – JustKitten
    Feb 14, 2018 at 18:46
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    $\begingroup$ Also note that $2\pi=360^\circ$ numerically is not true. It's the ratios that matter. $\endgroup$
    – JustKitten
    Feb 14, 2018 at 18:49
  • $\begingroup$ Ok. So $2\pi=360^\circ$ is always true. I guess I can't understand why because I don't know how a radian is defined, but I'll look it up. Thanks! $\endgroup$
    – Hanlon
    Feb 14, 2018 at 19:09
  • $\begingroup$ sorry, I don't understand that result you have written to find $d$, I have obtained another result: $$\begin{array}{lcl} \frac{180}{d} & = & \frac{\pi}{r} \\ \frac{d}{180} & = & \frac{r}{\pi} \\ \frac{d}{180} \cdot 180 & = & \frac{r}{\pi} \cdot 180 \\ d & = & \frac{180}{\pi}\cdot r \end{array}$$ Why I obtain this different result? thanks. $\endgroup$
    – JB-Franco
    Apr 19, 2018 at 11:08
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A full circle is $360^\circ$ and also $2\pi$ radians. Thus $360^\circ = 2\pi\text{ rad}$. We simplify by dividing by two.

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  • $\begingroup$ another case where τ would be more natural than π $\endgroup$
    – Henry
    Feb 14, 2018 at 17:35
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Think of it as solving proportions. We have $$\pi \text{ radians} = 180^\circ$$ and you want to convert $$r \text{ radians} = d^\circ.$$ Hence you get $$ \frac{\pi}{r} = \frac{180}{d} $$ which simplifies to $$ r = \frac{\pi}{180} d \quad \text{or equivalently} \quad d = \frac{180}{\pi} r. $$

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  • $\begingroup$ What if radius isn't 1? For example, if $r=2$, do I use $\frac{\pi}{90}$? I think yes, but just to confirm. $\endgroup$
    – Hanlon
    Feb 14, 2018 at 17:36
  • $\begingroup$ @Vuk nothing here depends on radius. Radians are a measure of angle not length $\endgroup$
    – gt6989b
    Feb 14, 2018 at 17:39
  • $\begingroup$ But the formula of circle's circumference is $r2\pi$. If $r=2$, then the circle's circumference ($360^\circ$) is $4\pi$. Then $90^\circ=\pi,180^\circ=2\pi$ and so on. I can't apply $\frac{\pi}{180}$, because $180^\circ \neq \pi$. $\endgroup$
    – Hanlon
    Feb 14, 2018 at 17:43
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The radian is defined as the plane angle subtended by any circular arc divided by its radius.

When the circular arc is actually congruent to the circle, the length is $2\pi r=2\pi=$ $\tau$ (for a unit circle). The angle subtended by this arc is $360^\text{o}$, and therefore $1\:\text{radian}=\frac{360}{\tau}=\frac{180}{\pi}$.

So: $$r\text{ radians}=\text{d}\cdot \frac{180}{\pi}\\ d\text{ degrees} = \frac{r}{\frac{180}{\pi}}=\frac{180r}{\pi}$$

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Since circumference =2pier if we take half circle it would be pier in which rotation of angle is 180 degree and arc length would be pie*r then we get theta =pie =180 degree

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