Why do we use $\frac{\pi}{180}$ to convert from degrees to radians?

Question

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?

Answer 1

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.

Answer 2

A full circle is $360^\circ$ and also $2\pi$ radians. Thus $360^\circ = 2\pi\text{ rad}$. We simplify by dividing by two.

Answer 3

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 $$ r = \frac{\pi}{180} d \quad \text{or equivalently} \quad d = \frac{180}{\pi} r. $$

Answer 4

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

Answer 5

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