Converting Between Degrees and Radians

Degrees cut a circle into 360 equal slices, an ancient, arbitrary choice. Radians measure an angle by how far you travel around the rim of a circle, using the radius itself as the ruler. Take the radius and lay it along the circle's edge, over and over. It takes exactly 2 π copies of the radius to go all the way around, because the circumference of a circle is 2 π r. So a full turn is 2 π radians, a half turn is π radians, and π radians lines up exactly with 180 degrees. That one fact, π radians = 180 degrees, is the whole conversion rule.

To go from degrees to radians, multiply by π over 180. To go from radians to degrees, multiply by 180 over π. For 135 degrees: 135 times π over 180 simplifies to 3 π over 4 radians. For 7 π over 6 radians: 7 π over 6 times 180 over π gives 7 times 180 over 6, which is 210 degrees. The π's cancel because you're converting units, not evaluating π.

This conversion matters because radians are the natural unit for right-triangle trig and circles: arc length, sector area, and the trig functions used in calculus all read more simply in radians. Degrees stay useful for everyday angle sense, like describing a 45 degree turn. Expect radians to show up any time a problem talks about arc length, rotation, or uses π directly in an angle measure.