Source: Unit 6c Unit Circle Packet, Texas A&M University
Tags: radians, radian measure, arc length, unit circle angles, degrees to radians, standard position, terminal ray, positive angle, negative angle, counterclockwise, clockwise
A radian measures an angle by comparing the arc length it cuts out to the radius of the circle. On the unit circle (radius = 1), the radian measure equals the arc length itself. Positive angles go counterclockwise from the positive x-axis; negative angles go clockwise.
Radian measure
The ratio of the arc length created by an angle to the radius of the circle. In other words, it tells you how far a point has travelled along the arc. Formula: θ = arc length / radius.
Standard position
An angle whose vertex sits at the origin and whose initial ray lies along the positive x-axis.
Terminal ray
The ray that rotates away from the initial ray (the positive x-axis) to form the angle.
Positive angle
An angle measured in the counterclockwise direction from the positive x-axis.
Negative angle
An angle measured in the clockwise direction from the positive x-axis.
Arc length
The distance along the curved edge of a circle between two points. Found by taking the relevant proportion of the full circumference: arc length = (fraction of full rotation) × 2πr.
Degree
An alternative angle unit that divides a full circle into 360 equal parts.
A radian is defined as θ = (arc length) / (radius).
This means that if the arc length equals the radius, the angle is exactly 1 radian.
Radians are dimensionless, since you are dividing a length by a length.
This ratio makes radians particularly useful for calculating distances along curved surfaces.
The circumference of a full circle is C = 2πr.
To find the arc length for a given fraction of the circle, multiply that fraction by 2πr.
Example: a quarter circle (1/4 of a full rotation) on a circle of radius 4 has arc length = (1/4)(2π)(4) = 2π.
The radian measure for that quarter-circle angle is then 2π / 4 = π/2.
A full rotation (360°) around a circle of radius r sweeps an arc length equal to the entire circumference: 2πr.
θ = 2πr / r = 2π radians.
So one full counterclockwise revolution = 2π radians.
Counterclockwise rotation from the positive x-axis produces a positive angle.
Clockwise rotation from the positive x-axis produces a negative angle.
The sign tells you direction only; the magnitude tells you how far.
8 equal pieces (multiples of π/4)
Each piece subtends an angle of 2π / 8 = π/4 radians.
Going counterclockwise from 0: π/4, 2π/4 = π/2, 3π/4, 4π/4 = π, 5π/4, 6π/4 = 3π/2, 7π/4, 8π/4 = 2π.
12 equal pieces (multiples of π/6)
Each piece subtends an angle of 2π / 12 = π/6 radians.
Going counterclockwise from 0: π/6, 2π/6 = π/3, 3π/6 = π/2, 4π/6 = 2π/3, 5π/6, 6π/6 = π, 7π/6, 8π/6 = 4π/3, 9π/6 = 3π/2, 10π/6 = 5π/3, 11π/6, 12π/6 = 2π.
1/6 of a circle clockwise = -(1/6)(2π) = -π/3 radians.
3/4 of a circle counterclockwise = (3/4)(2π) = 3π/2 radians.
2 full revolutions clockwise = -(2)(2π) = -4π radians.
Quadrant I: 0 < θ < π/2
Quadrant II: π/2 < θ < π
Quadrant III: π < θ < 3π/2
Quadrant IV: 3π/2 < θ < 2π
Radian measure: θ = arc length / radius
Arc length: s = rθ (where θ is in radians)
Full circumference: C = 2πr
Degree-radian conversions: 360° = 2π rad, so 180° = π rad. To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π.
⚠️ On the unit circle, the radian measure is numerically equal to the arc length, because r = 1. This simplification is the whole reason the unit circle is so useful.
⚠️ Don't confuse the sign convention: counterclockwise = positive, clockwise = negative. AP-style questions test this directly.
⚠️ When asked "which quadrant," convert the angle to see where it falls relative to π/2, π, 3π/2, and 2π. Common trap: 5π/6 is in Quadrant II (between π/2 and π), not Quadrant III.
⚠️ Memorise the key angle families: multiples of π/6 and multiples of π/4 appear constantly.
Q: What is the radian measure of a quarter-circle angle on a circle of radius 4?
A: Arc length = (1/4)(2π)(4) = 2π. Radian measure = 2π / 4 = π/2.
Q: What is the radian measure of a full counterclockwise rotation on any circle?
A: 2π radians, regardless of the radius.
Q: 1/6 of a circle in the clockwise direction is how many radians?
A: -π/3 radians (negative because clockwise).
Q: If point P is in Quadrant II, which could be θ: π/4, 5π/6, 4π/3, or 7π/4?
A: 5π/6. It falls between π/2 and π, which is the Quadrant II range.
Q: If point P is in Quadrant III, which could be θ: π/3, 3π/2, 5π/4, or 11π/6?
A: 5π/4. It falls between π and 3π/2. (Note: 3π/2 itself is the negative y-axis, not in any quadrant.)
Q: If point P is in Quadrant IV, which could be θ: π/6, π/2, 5π/6, or 5π/3?
A: 5π/3. It falls between 3π/2 and 2π.
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