AI Tutor
If you walk along the curved edge of a circular running track, how far did you actually travel without having to divide by 360∘360^\circ every single time?
Radians were invented specifically to solve this problem by directly linking the angle at the center of a circle to the distance along its edge.

The Arc Length Formula

An arc length (ss) is the curved distance along the circumference of a circle subtended by a central angle (ΞΈ\theta), which is an angle whose vertex sits at the circle's center.
Because a full circle covers 2Ο€2\pi radians and has a circumference of 2Ο€r2\pi r, a fraction of the circle ΞΈ2Ο€\frac{\theta}{2\pi} gives an arc length of s=(ΞΈ2Ο€)(2Ο€r)=rΞΈs = \left(\frac{\theta}{2\pi}\right)(2\pi r) = r\theta.
Let's find the arc length on a circle of radius r=6Β cmr = 6\text{ cm} intercepted by a central angle of ΞΈ=2Ο€3Β radians\theta = \frac{2\pi}{3}\text{ radians}.
Applying s=rΞΈs = r\theta, we multiply: s=6β‹…2Ο€3=4Ο€β‰ˆ12.57Β cms = 6 \cdot \frac{2\pi}{3} = 4\pi \approx 12.57\text{ cm}
Measuring the curved edge is straightforward, but what if you need to calculate the entire two-dimensional area of that pizza slice?

Sector Area

A sector is the region inside a circle bounded by two radii and their intercepted arc.
Since the total area of a circle is Ο€r2\pi r^2, a sector with central angle ΞΈ\theta takes up a proportion of ΞΈ2Ο€\frac{\theta}{2\pi}, giving the formula: A=(ΞΈ2Ο€)(Ο€r2)=12r2ΞΈA = \left(\frac{\theta}{2\pi}\right)(\pi r^2) = \frac{1}{2}r^2\theta