At its core, an angle is a measure of a turn. Think about opening a door or looking up at the top of a building. The amount you have to turn your head or swing the door creates an angle. Formally, an angle is formed by two rays that share a common endpoint.

The common endpoint is called the vertex, and the two rays are the sides of the angle. We can think of an angle as being formed by rotating one ray (the terminal side) away from another fixed ray (the initial side).

The direction of rotation matters. A counter-clockwise turn creates a positive angle, while a clockwise turn creates a negative angle. To talk about the size of an angle, we need a unit of measurement. The two most common units are degrees and radians.

Degrees are the most familiar way to measure angles. Imagine a full circle. If you divide that circle into 360 equal slices, each slice represents one degree. A small symbol, °, is used to denote degrees.

Why 360? The number comes from ancient Babylonians, who used a base-60 number system. 360 is conveniently divisible by many numbers (2, 3, 4, 5, 6, 8, 9, 10, 12, etc.), which made calculations easier for them. Certain angles have special names:

In higher-level math and physics, radians are often preferred over degrees. The definition of a radian is directly linked to the properties of a circle. One radian is the angle created when the arc length along the circle is equal to the length of the circle's radius.

This might seem a bit abstract, but it creates a very natural relationship. The size of a radian doesn't depend on an arbitrary number like 360; it's based on the circle itself.

So how many radians are in a full circle? The circumference of a circle is given by the formula $C = 2\pi r$. This means we can fit $2\pi$ (about 6.28) radius lengths along the full circumference. Therefore, a full 360° rotation is equal to $2\pi$ radians.

This simple relationship, $180^{\circ} = \pi$ radians, is the key to converting between the two systems.

To convert from one unit to another, you multiply by a conversion factor. Since $180^{\circ} = \pi$ radians, we can create two fractions that equal 1:

To convert degrees to radians, multiply by $\frac{\pi}{180^{\circ}}$. The degree units will cancel out, leaving you with radians.

For example, let's convert 60° to radians:

To convert radians to degrees, you do the opposite: multiply by $\frac{180^{\circ}}{\pi}$.

Let's convert $\frac{\pi}{4}$ radians to degrees:

Here's a quick reference for some common angles:

Let's review these key ideas before we move on.

Now, try a few practice problems to check your understanding.

Being comfortable with both degrees and radians, and especially how to convert between them, is a crucial first step in mastering trigonometry.