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.
One 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.
To convert between degrees and radians, we use the following relationships:
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°
When working with radians, it's often more convenient to leave answers in terms of π rather than calculating decimal approximations.
Radians simplify formulas for calculating the area of a circular sector and the length of an arc.
The area of a sector with radius $r$ and angle $\theta$ in radians is given by:
$$ A = \frac{1}{2}r^2\theta $$
Calculate 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
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