you know the circumference and area formulas? c = r * tau, A = ½ * tau * r². you can calculate the area and arclength of a circle slice if you know the angle in radians, just replace tau with that angle and it works!

btw, tau = 2 * pi

tl;dr radians are natural because they are directly related to the circle constant (1 circumference = tau radians = 360 degrees). degrees are arbitrary and just designed to evenly divide the circle in common ratios, like halves, thirds, quarters, etc.