Trigonometry
Why 180 Degrees Equals Pi Radians
Radian measure is arc length divided by radius. A half-circle arc has length pi times the radius, so its angle is pi radians; the same half turn is 180 degrees.
Reviewed
Radians measure an arc in units of the radius
A semicircle has arc length pi times its radius. Dividing that arc by the radius leaves pi radians.
Start with a circle's circumference
A circle of radius r has circumference 2 pi r. Traveling once around the circle traces an arc of length 2 pi r, so the angle at the center measures (2 pi r)/r = 2 pi radians.
Degree measure assigns 360 degrees to that same full turn. Halving both descriptions gives pi radians and 180 degrees.
theta = arc length / radius; (pi r)/r = pi radians = 180 degreesDerive the equality from one full circle
No decimal approximation of pi is needed.
C = 2pi rA full circle has circumference 2pi times its radius.theta = C/r = 2piThe full-turn angle in radians is arc length divided by radius.half-turn = pi radiansA semicircle is half of the full turn.half-turn = 180 degreesThe degree definition assigns 360 degrees to a full turn.180 degrees = pi radiansBoth expressions name the same half-turn.
The radius cancels
The relationship does not depend on circle size. Doubling the radius doubles the corresponding arc length, leaving their ratio unchanged.
That invariance is why radians describe the angle itself rather than a particular circle.
Worked example 2
Derive 45 degrees
A 45-degree angle is one quarter of a half turn.
180 degrees / 4 = 45 degreespi radians / 4 = pi/4 radians
Result: 45 degrees = pi/4 radians
Questions about why 180 degrees equals pi radians
Does pi radians mean pi times a unit?
It is an exact radian measure. The named unit clarifies that the number describes an angle.