Understand radian measure as arc length on the unit circle and convert between radians and degrees

Learning goal: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle, convert between radians and degrees, and use the relationship s = rθ to find arc lengths on circles of any radius.

Two ways to measure an angle. Degrees are a convention: someone decided that a full turn is 360 equal parts. Nothing in the geometry forces the number 360. Radian measure is different, because it is built from the circle itself. Draw a circle of radius 1, called the unit circle, and place the vertex of an angle at its center. The two sides of the angle cut off a piece of the circle, an arc; we say the angle subtends that arc. The radian measure of the angle is the length of that arc. An angle of 1 radian subtends an arc of length exactly 1 on the unit circle, that is, an arc as long as the radius. An angle of 2 radians subtends an arc of length 2, and so on. On a circle of any radius the radian measure is the ratio of arc length to radius, and on the unit circle that ratio is simply the arc length. A length divided by a length is a pure number, so radian measure is dimensionless: the radian is a named unit, but it cancels in calculations like s = rθ; that is why the word "radians" is often dropped and an angle is written simply as 2 or as π/3.

Why π appears. The circumference of the unit circle is 2π × 1 = 2π. A full turn subtends the whole circle, so a full turn measures 2π radians. Half a turn, which is 180°, subtends half the circumference, so 180° = π radians. This single equation is the conversion fact from which everything else follows. Dividing both sides by 180 gives 1° = π/180 radians ≈ 0.01745 radians. Dividing both sides by π gives 1 radian = 180°/π ≈ 57.30°. A radian is a large angle, a little less than 60°, and a little more than six of them (2π ≈ 6.283) make a full turn.

Converting. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. Keep π in the answer when the degree measure is a nice fraction of 180°, because the exact form is what later lessons use: 60° = 60 × π/180 = π/3; 135° = 135π/180 = 3π/4; 270° = 270π/180 = 3π/2. Going the other way, 5π/6 = (5/6)(180°) = 150° and 7π/4 = (7/4)(180°) = 315°. When the radian measure is not a multiple of π, use a decimal: 2 radians = 2 × 180°/π ≈ 114.59°, and 1.5 radians ≈ 85.94°.

A fully worked model. Convert 225° to radians, then check. 225 × π/180 = 225π/180. Both 225 and 180 are divisible by 45: 225 ÷ 45 = 5 and 180 ÷ 45 = 4, so the angle is 5π/4 radians. Check by going back: 5π/4 × 180°/π = 5 × 45° = 225°. Check the size as well: 5π/4 ≈ 3.93, which is more than π ≈ 3.14 (a half turn) and less than 3π/2 ≈ 4.71 (three quarters of a turn), and 225° is indeed between 180° and 270°.

Arcs on circles of any radius. The unit circle defines radian measure, but the same angle sits at the center of circles of every radius, and all of those circles are scaled copies of the unit circle. If the radius is multiplied by r, every length in the figure is multiplied by r, including the arc. So an angle of θ radians at the center of a circle of radius r subtends an arc of length s = rθ. This formula is only correct when θ is in radians. On a circle of radius 6 cm, an angle of 2π/3 subtends an arc of length 6 × 2π/3 = 4π ≈ 12.57 cm. If a student put 120° into the formula they would get 720 cm, which is longer than the whole circumference of about 37.7 cm; that nonsense answer is the signal that a degree measure was used where a radian measure was required. The formula also runs backwards: an arc of 12 m on a circle of radius 8 m subtends a central angle of 12/8 = 1.5 radians, about 85.94°.

The misconception. Students often think "radians are just degrees divided by 57.3", a second, more awkward unit for the same thing. The conversion factor is real, but it misses the point. Radian measure says what the angle does to the circle: it is the arc length per unit of radius. That is why s = rθ has no extra constant in it, why the circumference formula 2πr is the special case θ = 2π, and why the trigonometric functions in the rest of this unit take radian inputs. When a wheel of radius 0.35 m turns through 100 radians, a point on its rim travels 0.35 × 100 = 35 m, and the same statement in degrees would need a factor of π/180 every time.

Reading a radian measure on the unit circle. To locate the angle 2 on the unit circle, start at the point (1, 0), move counterclockwise along the circle a distance of 2, and stop. Because a quarter turn is π/2 ≈ 1.571 and a half turn is π ≈ 3.142, the angle 2 ends in the second quarter of the circle. The angle 5 ends in the fourth quarter, because 3π/2 ≈ 4.712 < 5 < 2π ≈ 6.283. Estimating where an angle lands is a skill worth practicing before the next lesson, which uses these positions to define sine and cosine for every real number.

Worked conversion, degrees to radians: Convert 150° to radians. 150 × π/180 = 150π/180. Divide numerator and denominator by 30: 5π/6. Check the size: 5π/6 ≈ 2.618, between π/2 ≈ 1.571 and π ≈ 3.142, just as 150° lies between 90° and 180°.

Worked conversion, radians to degrees: Convert 11π/6 to degrees. 11π/6 × 180°/π = 11 × 30° = 330°. Also express 3 radians in degrees: 3 × 180°/π ≈ 171.89°. Notice 3 radians is a little less than a half turn (π ≈ 3.142), which agrees with 171.89° being a little less than 180°.

Worked arc length: The minute hand of a clock is 12 cm long. How far does its tip travel in 25 minutes? In 60 minutes the hand turns 2π radians, so in 25 minutes it turns 25/60 × 2π = 5π/6 ≈ 2.618 radians. The tip moves on a circle of radius 12 cm, so s = rθ = 12 × 5π/6 = 10π ≈ 31.42 cm. In degrees the hand turned 150°, and the ratio 150/360 = 5/12 of the circumference 24π gives the same 10π cm, which confirms the formula.

Worked central angle from an arc: A circular running track has radius 40 m. A runner covers 60 m of the curve. Through what angle at the center did the runner turn? θ = s/r = 60/40 = 1.5 radians, which is 1.5 × 180°/π ≈ 85.94°. The answer is a pure number, not "1.5 meters": the meters cancel in s/r, which is exactly why radian measure is dimensionless and the unit name radian can be dropped.

Representation task: Draw a unit circle with radius 6 cm on paper (so 1 unit = 6 cm). Mark the angles 0, π/6, π/4, π/3, π/2, π, 3π/2 and 2π on it by measuring their degree equivalents (0°, 30°, 45°, 60°, 90°, 180°, 270°, 360°) with a protractor. Wrap a piece of string along the arc from (1, 0) to the π/3 mark, straighten it and measure it; record that it is close to 6 × π/3 ≈ 6.28 cm, that is, a little more than one radius (π/3 ≈ 1.047 radii). Record a two-column table of degree and radian measures for all eight angles.

Independent performance: Write a one-page explanation for a classmate who has never met radians. It must define radian measure as an arc length on the unit circle, explain why a full turn is 2π and a half turn is π, derive the conversion factors π/180 and 180/π from 180° = π, convert three angles you choose in each direction (at least one that is not a multiple of 30° or 45°) and solve one arc-length problem on a circle whose radius is not 1, showing s = rθ and stating the units of the answer. Success criteria: every conversion is correct when checked by reversing it; the arc-length answer carries the unit of the radius and is smaller than the circumference when the angle is less than one full turn; and the explanation says why θ must be in radians for s = rθ to work.