Using Degrees vs. Radians in Geometry

Radians are a different way to measure angles. You may be more familiar with its more commonly used cousin, degrees.

Why do mathematicians prefer radians over degrees? Think about it this way.

In fact, there is another, perhaps obscure, way to measure angles. This unit is called the gradian, and it is defined as $1/400$ths of a full circle.

However, a radian is defined more concretely. Imagine a circle with radius $r$:

We take the radius of the circle, and wrap it around the circumference once:

The angle that the arc makes is defined as $1$ radian, or $1\text{ rad}$.

So a question that naturally occurs: how many radians are in a full circle? Well, let's try it out.

Hmm... we can fit slightly more than $3$ radians in a half of the circle.

So if a half of a full circle is $\pi\text{ rad}$, then that means a full circle is exactly $2\pi\text{ rad}$. Cool!

So now that we've established that a full circle is $2\pi$ radians and proved it, we can also create simple conversions between the radian and the degree.

Remember that $2\pi$ radians corresponds to $360^{\circ}$. So if we have $x$ radians, then to turn it into degrees (remember your Dimensional Analysis), we multiply by $\dfrac{360^{\circ}}{2\pi\text{ rad}}$. This way, the radians cancel out, and we are left with the number of degrees. We can also simplify our fraction, and our formula turns into: $$x \text{ rad} = \left(x\cdot \dfrac{180}{\pi}\right)^{\circ}$$

where we multiply our radian measure by $\dfrac{180}{\pi}$.

Similarly, we can find the rule for changing degrees into radians: $$x^{\circ} = \left(x\cdot \dfrac{\pi}{180}\right)\text{ rad}$$

where we multiply our degree measure by $\dfrac{\pi}{180}$.

Congrats! You have learned What is a radian?, and also How a radian is derived. Also, we learned how to convert between radians and degrees.

If you do not understand any part of this, go back and read through it again. Thanks for reading!

Radians are a form of angle measurement. Just like there are $360^{\circ}$ in one full rotation of a circle, there are $2\pi$ radians.

To go from degrees to radians, multiply the degree value by $\frac{\pi}{180}$.

To go from radians to degrees, multiply the radian value by $\frac{180}{\pi}$.

Why don't you try a few examples.

Great job!

A degree is a measure of angle equal to $\frac{1}{360}$ of a full rotation.

From the definition of radians, $2\pi$ radians make up a full rotation.

Using both of the above equations, $180$ degrees equals $\pi$ radians.

Thus, to convert from degrees to radians, multiply the number of degrees by $\frac{\pi}{180}$, and to convert from radians to degrees, multiply the number of radians by $\frac{180}{\pi}$.

So what is the benefit of using degrees or radians?

For most practical applications outside mathematics (e.g. architecture and geography), degrees are much easier to comprehend and use, since degrees are what everybody is accustomed to.

On the other hand, radians reign supreme in pure mathematics and physics. Radians have the following benefits:

They are dimensionless, which means that they can be treated just as numbers (although you still do not want to confuse Hertz with radians per second).

Radians give a very natural description of an angle (whereas the idea of $360$ degrees making a full rotation is very arbitrary).

The trigonometric functions have very simple derivatives when one uses radians, due to some nice limits. This leads to other benefits, such as convenient Maclaurin series expressions for the trigonometric functions.

Unfortunately, we cannot go around teaching little children angles in radians, so we have to teach them in degrees first and introduce radians. However, the conversion is fairly simple, anyway. The trick is to always double-check which unit of angle measure you are working with.

Here is a worksheet for practicing your new skills on degrees vs. radians (see Sujay Kazi's explanation first):