Radian measure (as ratio of arc length to radius); degree-radian conversion
Radian Measure
Radian Measure is a topic in Circles in the California Common Core State Standards. It is aligned to Standard G-C.5, which requires students to derive the definition of the radian measure of an angle as the constant of proportionality between arc length and radius.
Radian measure defines an angle as arc length divided by radius, so a straight angle is \(\pi\) radians and \(180^\circ=\pi\).
Theory
A radian is the central angle whose intercepted arc equals the radius. In general the radian measure is the ratio
Since the full circumference is \(2\pi r\), a full turn is \(2\pi\) radians \(=360^\circ\), so \(\pi\) radians \(=180^\circ\).
Definition, conversion, and arc length:
How to use radians
- Degrees to radians: multiply by \(\dfrac{\pi}{180}\).
- Radians to degrees: multiply by \(\dfrac{180}{\pi}\).
- As a ratio: \(\theta=\dfrac{s}{r}\).
- Arc length: \(s=r\theta\) (radians).
Multiply by \(\dfrac{\pi}{180^\circ}\).
| \(90^\circ\cdot\dfrac{\pi}{180^\circ}\) | \(=\) | \(\dfrac{\pi}{2}\) |
Multiply by \(\dfrac{180^\circ}{\pi}\).
| \(\dfrac{\pi}{6}\cdot\dfrac{180^\circ}{\pi}\) | \(=\) | \(30^\circ\) |
A radian measure is arc length over radius.
| \(\theta\) | \(=\) | \(\dfrac{15}{5}=3\ \text{rad}\) |
Use \(s=r\theta\) with \(\theta\) in radians.
| \(s\) | \(=\) | \(12\cdot\dfrac{\pi}{3}=4\pi\) |
Common pitfalls
Frequently asked questions
What is a radian?
The central angle whose intercepted arc length equals the radius; in general it is arc length divided by radius.
How many radians are in a full circle?
\(2\pi\) radians, which equals \(360^\circ\).
How do you convert between degrees and radians?
Degrees to radians: multiply by \(\pi/180\). Radians to degrees: multiply by \(180/\pi\).
What is the arc-length formula in radians?
\(s=r\theta\), with \(\theta\) in radians.