ACT® · Math: Functions — Trigonometric Functions
Radians as Arc Length on the Unit Circle
A radian is the angle made when you wrap the radius of a circle along its own edge. On the unit circle, that angle equals the arc length it cuts off, which is why radians make arc length and area formulas clean.
Degrees chop a circle into 360 equal slices for no reason tied to the circle itself, just old counting habits. Radians measure the angle by comparing it to the radius. Take the radius, a straight piece of string the length from center to edge, and lay it along the circle's rim. The angle you sweep out doing that is one radian. Since the circle's full rim length is 2 π times the radius, a full turn takes 2 π of those radius-lengths to wrap around, so a full turn is 2 π radians instead of 360 degrees.
That one definition makes radians the natural unit the moment a circle is involved. On the unit circle, radius 1, the arc length cut off by an angle is just the radian measure itself, no extra multiplying needed. That is the whole payoff: arc length s on any circle is r times theta, but only when theta is in radians. Degrees carry a hidden leftover factor of π/180 in every such formula, which is why swapping π straight in for 180 breaks things, π is a number near 3.14, not a stand-in symbol for the word degrees.
This sits right before the sine wave traced by a spinning point, since that wave is plotted against radian angle, not degrees, so the horizontal axis ticks in multiples of π. It also feeds directly into sector area and arc length problems, and into every identity written with π/6, π/4, π/3 instead of 30, 45, 60.
Key terms
- radian
- The angle formed when the radius of a circle is laid along the circle's own edge.
- arc length
- The distance along the curved edge of a circle between two points, equal to r times theta in radians.
- unit circle
- A circle of radius 1 centered at the origin, where radian angle and arc length are numerically equal.
- sector
- A pie-slice region of a circle bounded by two radii and the arc between them.
Watch it explained
- How it worksHow does laying the radius along the circle's own edge define one radian?
- Worked exampleFind the radian measure of 135 degrees and the degree measure of 5pi over 6.
- Common mistakeWhy doesn't π simply stand in for 180 everywhere it appears in a formula?
- Going furtherWhat happens to arc length on a circle of radius 7 when the angle is given in radians?
Check yourself
End of topic.