Degrees and Radians

Question

If there are exactly $2\pi$ radians in a circle, then how do we derive the formula to convert from degrees to radians?

I understand that there are 360 degrees in a circle. Hence $2\pi$ radians = $360^{\circ}$ So 1 radian = $\frac{180}{\pi}$ degrees. Shouldn't this be the formula for converting from degrees to radians? Why is it radians = $\frac{\pi}{180}$ degrees?

Answer 1

If you regard $\,^\circ$ as meaning "multiplied by $\frac{2\pi}{360}$" i.e. by $\frac{\pi}{180}$, then

• An angle of $360^\circ = 2\pi$
• An angle of $180^\circ = \pi$
• An angle of $60^\circ = \frac\pi3 \approx 1.04719755$
• An angle of $1^\circ = \frac{\pi}{180} \approx 0.01745329$

with the right-hand values being radians.

If you want to go in the opposite direction so as to insert a $\,^\circ$ to give an answer in degrees, you have to divide the value (in radians) by $\frac{\pi}{180}$, i.e. multiply by $\frac{180}{\pi}$, so

• An angle of $1 =\left(\frac{180}{\pi}\right)^\circ\approx 57.29577951^\circ$
• An angle of $3 =\left(\frac{540}{\pi}\right)^\circ\approx 171.8873385^\circ$
• An angle of $2\pi =\left(2\pi \times\frac{ 180}{\pi}\right)^\circ=360^\circ$

Answer 2

You can use the formula $180^\circ = \pi\,rad$ as a formula for conversion. Use one of the following forms: $$\frac{180^\circ}{\pi\,rad} = 1$$ or $$\frac{\pi\,rad}{180^\circ} = 1$$ Since both of the ratios equal 1, multiplying an angle by these ratios does not change the angle, just the representation. The trick is to pick the one that cancels the right unit.

So, if you want $10^\circ$ in radians, then you use the formula with degrees in the denominator: $$10^\circ\frac{\pi\,rad}{180^\circ} = \pi/18\,rad$$ The degree units cancel, leaving only radians.

Answer 3

This is confusing to many people who first encounter units conversions, but the larger a unit is, the less of it you use for a given measurement.

Conversely, the smaller a unit is, the more of it you use for a given measurement.

In the formula $$ 1\ \mathrm{radian} = \frac{180}{\pi}\ \mathrm{degrees}, \tag1 $$ we are comparing the sizes of two different angles, one angle of $1$ radian and another of $1$ degree. The angle of $1$ radian is larger, specifically $180/\pi \approx 57.296$ times as large.

The other formula is not as self-explanatory. What does "radians" mean in an equation? It may be a little clearer if we write it like this, $$ (\#\mathrm{radians}) = \frac{\pi}{180}(\#\mathrm{degrees}), \tag2 $$ and explain that in this case we are comparing two different ways of measuring the same angle, whose measurement is $(\#\mathrm{radians})$ radians and is also $(\#\mathrm{degrees})$ degrees. Notice that there are no units in this equation, only the numbers $(\#\mathrm{radians})$ and $(\#\mathrm{degrees})$.

Equation $(2)$ works because if we have an angle of $57.296$ degrees, it should be about $1$ radian. If we were to take the number of degrees, which is $57.296,$ and multiply by $\frac{180}{\pi}$, we would get $$ \frac{180}{\pi} \times 57.296 \approx 57.296^2, $$ which is a lot larger than $1$ and obviously the wrong answer. But if we multiply the number of degrees by $\frac{\pi}{180}$ instead, we get $$ \frac{\pi}{180} \times 57.296 \approx 1, $$ that is, $57.296$ degrees is very nearly $1$ radian. This is consistent with Equation $(1)$, which told us that $1$ radian was very nearly $57.296$ degrees.

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Another way to look at it is, Equation $(1)$ leads directly to the equation $$ (\#\mathrm{degrees}) = \frac{180}{\pi}(\#\mathrm{radians}), \tag3 $$ which allows us to easily convert radians to degrees. So if you have $(\#\mathrm{radians}) = 1$ on the right, you get $(\#\mathrm{degrees}) = \frac{180}{\pi}$ on the left -- literally, $1$ radian is $\frac{180}{\pi}$ degrees.

If you have $(\#\mathrm{radians}) = 2$ on the right, you get $(\#\mathrm{degrees}) = \frac{360}{\pi}$ on the left. And if you have $(\#\mathrm{radians}) = 2\pi$ on the right, you get $(\#\mathrm{degrees}) = 360$ on the left, that is, $2\pi$ radians is a full circle.

If you compare Equations $(2)$ and $(3)$, you should find that by algebra they are exactly equivalent to each other.