Angle of Rotation
What is an Angle of Rotation?
An angle of rotation is the measure of the amount a ray (or line segment) has been rotated about a fixed point called the vertex or center of rotation.
Imagine the hands of a clock. As the minute hand moves from 12 to 3, it sweeps through a certain angle. That swept angle is the angle of rotation.
Formal Definition: An angle of rotation is formed by rotating an initial side (starting position) to a terminal side (ending position) about a fixed point called the vertex. The amount of rotation determines the measure of the angle.
Key Terminology
| Term | Definition |
|---|---|
| Initial Side | The starting position of the ray before rotation. In standard position, this lies along the positive $x$-axis. |
| Terminal Side | The ending position of the ray after rotation. |
| Vertex | The fixed point about which the rotation occurs. In standard position, this is the origin $(0, 0)$. |
| Positive Angle | Rotation in the counterclockwise direction. |
| Negative Angle | Rotation in the clockwise direction. |
| Standard Position | An angle whose vertex is at the origin and whose initial side lies along the positive $x$-axis. |
Measuring Angles of Rotation
Degrees
A full rotation = $360°$. Common rotation angles include:
- Quarter turn: $90°$
- Half turn: $180°$
- Three-quarter turn: $270°$
- Full turn: $360°$
Radians
A full rotation = $2\pi$ radians. The conversion formula is:
$$\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180°}$$
$$\theta_{\text{degrees}} = \theta_{\text{radians}} \times \frac{180°}{\pi}$$
Angles Beyond 360° and Coterminal Angles
An angle of rotation is not limited to values between $0°$ and $360°$. A ray can rotate more than one full turn!