Degrees and Radians

Pre-Calculus · Axiom Academy

Two ways to measure the same angle — and why radians come straight from the circle itself. Take the radius of a circle and bend it around the rim. The angle it sweeps out at the center — the angle whose arc length equals the radius — is exactly one radian . In the animation the straight radius (orange) unrolls onto the arc until the arc is one radius long; that opening is 1 rad. How many radius-lengths fit around a full circle? The circumference is , and each radian is one radius of arc — so a full turn is exactly radians , the same sweep as . Walk only halfway around and you've laid down radius-lengths: that half turn is . The animation swings a pointer from to , dropping radius-length ticks (orange) along the way. so half of that — the most important conversion 3. Converting Between Degrees and Radians Because radians, one degree is of a radian, and one radian is degrees. To convert, multiply by whichever ratio cancels the unit you're leaving. The animation feeds through the machine and the angle settles at on the circle. Convert to radians: radians. Going back: . The 's and the numbers cancel cleanly — that's the sign you used the right ratio. Degrees are the "big" numbers and radians the "small" ones (a right angle is but only rad). So going degrees → radians you shrink — multiply by . Going the other way you grow — multiply by .

This is the written version of the interactive lesson above. See the full Pre-Calculus course.