a radian
/ RAY-dee-an /
We measure angles in degrees out of habit — 360 of them in a full turn — but 360 is an arbitrary number inherited from ancient astronomy. The radian is the natural unit nature actually prefers, defined not by a count but by the geometry of the circle itself, and it is the unit in which all higher mathematics measures angles.
Wrap the radius of a circle along its own rim, like laying a piece of string of length r around the edge. The angle at the centre that this arc subtends is exactly one radian. Equivalently, an angle in radians is arc length divided by radius, theta = s / r, which makes it a pure ratio with no units. Because the full circumference is 2 x pi x r, a whole turn is 2 x pi x r divided by r = 2 x pi radians; so 360 degrees = 2 x pi radians, and 180 degrees = pi radians. To convert, multiply degrees by pi / 180 to get radians, or multiply radians by 180 / pi to get degrees. So 90 degrees = pi/2, 60 degrees = pi/3, 45 degrees = pi/4, 30 degrees = pi/6.
Radians pay off twice. First, arc length and sector area become effortless: an arc is s = r x theta and a sector's area is A = (1/2) x r^2 x theta, but ONLY when theta is in radians. Second, the calculus of trig functions is clean in radians (the derivative of sin theta is cos theta) and ugly in degrees. Calculators have a degree/radian mode switch precisely because mixing them is the most common silent error in trigonometry.
A circle has radius 6 cm. A central angle of 1.5 radians cuts off an arc of length s = r x theta = 6 x 1.5 = 9 cm, and the sector it bounds has area A = (1/2) x r^2 x theta = (1/2) x 36 x 1.5 = 27 cm^2. Both formulas require radians; in degrees they would be wrong.
Arc length r x theta and sector area (1/2) r^2 theta are clean only when the angle is in radians.
The arc-length and sector formulas s = r theta and A = (1/2) r^2 theta hold ONLY for theta in radians; plug in degrees and you get a meaningless number — convert first.