An angle is what you get when two lines, or rays, meet at a single point. Think of the hands on a clock. The point where they join in the center is called the vertex, and the hands themselves are the rays.

The size of an angle measures the amount of turn or “opening” between the two rays. The longer the rays are, it doesn't mean the angle is bigger. The angle is all about the rotation from one ray to the other.

The most common way to measure an angle is in degrees. Imagine a full circle. That complete rotation is divided into 360 equal parts, and each part is one degree ($1^\text{o}$). A half-circle is $180^\text{o}$, and a quarter-turn is $90^\text{o}$.

Angles are often classified by their size:

A right angle is exactly $90^\text{o}$, like the corner of a square.

An acute angle is any angle less than $90^\text{o}$. You can remember it as a "cute" little angle.

An obtuse angle is greater than $90^\text{o}$ but less than $180^\text{o}$.

Degrees are useful, but in many areas of math and physics, it's better to use a different unit: the radian. Radians relate an angle directly to the properties of a circle.

This creates a natural, unit-less way to measure angles. How many of these radius-length arcs fit around a whole circle? The circumference of a circle is $2\text{π}r$, so exactly $2\text{π}$ of them fit. That gives us the most important conversion:

Since we often work with simpler fractions of a circle, it's also useful to know that $180^\text{o} = \text{π}$ radians.

Switching between degrees and radians is a key skill. It all comes down to the relationship $180^\text{o} = \text{π}$ radians. To convert, you just multiply by a fraction that equals one, so you don't change the value of the angle.

Let's try an example. How many radians is $60^\text{o}$? We multiply by the conversion factor to cancel out the degree units.

Now let's go the other way. How many degrees is $\frac{\pi}{2}$ radians? This time, we want to cancel the $\pi$ and the radian unit.

You'll see this is a right angle, which makes sense since $\frac{\pi}{2}$ is one-quarter of a full circle's $2\pi$ radians.