An angle measures the amount of turn between two lines, or rays, that share a common endpoint. This shared point is called the vertex.

Think of an angle as a story of rotation. We start with an initial side, which is a ray sitting in a fixed position. Then, we rotate another ray, called the terminal side, away from the initial side. The amount of rotation between these two rays is the angle.

By convention, we measure angles starting from an initial side that lies on the positive x-axis. A counter-clockwise rotation creates a positive angle, while a clockwise rotation creates a negative angle.

We have two common ways to measure angles: degrees and radians. You're probably most familiar with degrees.

A full circle is divided into 360 degrees, written as $360^\circ$. This number is convenient because it's divisible by many other numbers, making it easy to talk about fractions of a circle.

• A $90^\circ$ angle is a quarter turn, forming a right angle.
• A $180^\circ$ angle is a half turn, forming a straight line.
• A $360^\circ$ angle is a full turn, bringing you back to where you started.

The other unit, radians, is fundamental in higher math like calculus. A radian relates the angle directly to the radius and arc length of a circle.

One radian is the angle created when the arc length is equal to the radius of the circle. Since the circumference of a circle is $2\pi r$, a full $360^\circ$ rotation is equal to $2\pi$ radians.

Knowing that $180^\circ = \pi$ radians gives us a straightforward way to convert between the two units. We can create two conversion factors that are equal to 1:

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

For example, let's convert $60^\circ$ to radians:

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

Let's convert $\frac{3\pi}{4}$ radians to degrees:

Here are some of the most common angles you'll encounter.

Understanding angles is the first step into trigonometry. These measurements form the input for trigonometric functions, which help us relate angles to side lengths in triangles and model repeating patterns in the world around us, like sound waves and orbits.