Radian Measure

Algebra 2 · Axiom Academy

The angle whose arc equals the radius — where, on the unit circle, an angle in radians is simply the arc length it sweeps. Take the radius of a circle and bend that exact length along the rim. The angle it opens at the center is one radian . Since the arc and the radius are the same length, the angle is just their ratio s/r — and one radian works out to about . Radians = arc length divided by radius When the arc equals the radius, the angle is exactly 1 radian The whole rim of a circle has length . Laying the radius-length arc end to end around the circle, it takes about 6.28 copies to come all the way back — and that number is exactly . So a full turn is radians , and a half turn is radians . 3. Converting Between Degrees and Radians Because radians, you convert by scaling. Multiply degrees by to get radians; multiply radians by to go back. On any circle, the arc cut off by a central angle is — as long as is in radians. Grow the angle and the arc grows right with it. This is the payoff of radians: measure the angle any other way and the formula picks up an extra conversion factor. With radius r = 3 and angle radians, the arc length is: You have seen a radian as arc length turned into an angle — and why that makes , the conversions, and fall out so cleanly. Scroll up to revisit any step.

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