Definition of a Radian
A radian is a unit of angular measure defined as the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. This definition provides a natural and fundamental way to measure angles, particularly useful in advanced mathematics and physics.
NoteOne radian is approximately equal to 57.2958 degrees.
The full circle, which measures 360° in degrees, is equal to $2\pi$ radians. This relationship forms the basis for conversion between degrees and radians.
Conversion Between Degrees and Radians
To convert between degrees and radians, we use the following relationships:
- 1 radian = $\frac{180°}{\pi}$ ≈ 57.2958°
- 1 degree = $\frac{\pi}{180}$ radians ≈ 0.0175 radians
To convert 45° to radians: 45° × $\frac{\pi}{180}$ = $\frac{\pi}{4}$ radians
To convert $\frac{\pi}{3}$ radians to degrees: $\frac{\pi}{3}$ × $\frac{180°}{\pi}$ = 60°
TipWhen working with radians, it's often more convenient to leave answers in terms of π rather than calculating decimal approximations.
Calculating Area of Sector and Length of Arc
Radians simplify formulas for calculating the area of a circular sector and the length of an arc.
Area of a Sector
The area of a sector with radius $r$ and angle $\theta$ in radians is given by:
$$ A = \frac{1}{2}r^2\theta $$
ExampleCalculate the area of a sector with radius 5 cm and central angle $\frac{\pi}{3}$ radians.
$A = \frac{1}{2} × 5^2 × \frac{\pi}{3} = \frac{25\pi}{6}$ cm²
Length of an Arc
The length of an arc $s$ with radius $r$ and angle $\theta$ in radians is:
$$ s = r\theta $$
ExampleFind the length of an arc with radius 10 m and central angle $\frac{\pi}{4}$ radians.
$s = 10 × \frac{\pi}{4} = \frac{5\pi}{2}$ m
Common MistakeStudents often forget to use radians in these formulas. If given an angle in degrees, always convert to radians first!
Expressing Radian Measure
Radian measures can be expressed in several forms:
- Exact multiples of π (e.g., $\frac{\pi}{6}$, $\frac{5\pi}{4}$)
