how to derive an arc leght and area of a sector from the formulas of a circle?

Question

also what is the relationship between them?

Answer 1

The circumference of a circle is 2pir
A central angle, x, is an angle between two radii.
The arc intercepted by the central angle will have a length equal to x/360 *2pir = pirx/180 if x is measured in degrees.
One radian = 180/pi degrees. So if x is measured in radians, arc length is simply rx.

The area of a circle is pir^2
So area of a sector x/360 *pir^2 if x measured in degrees, and is xr^2/2 if x measured in radians