Find arc lengths and sector areas, define radian measure as a ratio, and derive the formulas that convert between degrees and radians

Learning goal: Apply similarity to relate the length of an arc intercepted by a central angle to the radius, use that relationship to define the radian measure of an angle and to solve problems; derive the formula for the area of a sector and use it to solve problems; and derive the formulas that relate degrees and radians and convert between the two.

Arc length by proportion. The circumference of a circle of radius r is 2πr. A central angle of θ degrees cuts off an arc that is the fraction θ/360 of the whole circle, so the arc length is s = (θ/360)·2πr. The reasoning is proportional: the arc and the whole circle are parts of the same curve, and the central angle measures what fraction of a full turn the arc covers. For example, a 60° central angle in a circle of radius 9 intercepts an arc of length (60/360)·2π·9 = (1/6)·18π = 3π ≈ 9.42 units.

Apply similarity to relate arc length to radius. Here is the idea that turns arc length into a definition of angle. All circles are similar, and the dilation from a circle of radius r₁ to a concentric circle of radius r₂ has scale factor r₂/r₁; it maps the arc intercepted by a central angle θ on the small circle onto the arc intercepted by the same angle on the large circle, multiplying its length by r₂/r₁. So for a fixed central angle, arc length is proportional to radius: s₂/s₁ = r₂/r₁, or equivalently s₁/r₁ = s₂/r₂. In numbers: a 40° angle cuts an arc of (40/360)·2π·3 = 2π/3 ≈ 2.09 from a circle of radius 3 and an arc of (40/360)·2π·6 = 4π/3 ≈ 4.19 from a circle of radius 6; the ratio of arcs is 2, the ratio of radii, and the ratio s/r is 2π/9 ≈ 0.698 for both. That ratio depends only on the angle, not on the circle.

Radian measure defined. Because s/r is the same for every circle, it can serve as a measure of the angle. The radian measure of a central angle is the ratio of the length of the arc it intercepts to the radius: θ = s/r. It is a pure number (a length divided by a length), so it is dimensionless; the radian (rad) is still the name of the angle unit, and writing "rad" is useful to distinguish it from degrees. One radian is the angle whose arc equals the radius. A full turn intercepts the whole circumference, so a full turn is 2πr/r = 2π radians. Turning the definition around gives the arc-length formula in its cleanest form: s = rθ when θ is in radians. In a circle of radius 5, a central angle of 2.4 radians intercepts an arc of length 5·2.4 = 12; an arc of length 10 in a circle of radius 4 subtends an angle of 10/4 = 2.5 radians.

Derive the degree–radian formulas. A full turn is 360° and also 2π radians, so 360° = 2π radians and, dividing by 2, 180° = π radians. Dividing by 180 gives 1° = π/180 radians ≈ 0.01745 radians; dividing instead by π gives 1 radian = 180/π degrees ≈ 57.30°. Therefore: to convert degrees to radians, multiply by π/180; to convert radians to degrees, multiply by 180/π. The formulas are the two ways of writing one fact, that a straight angle is both 180° and π radians. Conversions: 150° = 150·π/180 = 5π/6 radians ≈ 2.618; 3π/4 radians = (3π/4)(180/π) = 135°; 2 radians = 2·180/π ≈ 114.59°; 2.5 radians ≈ 143.24°. Common landmarks worth knowing exactly: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π.

Derive the sector area formula. A sector is the region bounded by two radii and the arc between them, like a slice of pie. Its area is the same fraction of the whole circle's area that its central angle is of a full turn, because sectors with equal central angles are congruent, a full set of them fills the circle, and joining sectors adds both their angles and their areas, so area is proportional to angle. With the angle θ in degrees, A = (θ/360)·πr². With θ in radians the fraction of a full turn is θ/(2π), so A = (θ/(2π))·πr² = ½r²θ. A second derivation shows why the formula looks like a triangle's: cut the sector into many thin slivers; each is nearly a triangle with height r and a tiny base along the arc, so the total area is about ½·r·(sum of the bases) = ½·r·s = ½·r·(rθ) = ½r²θ. For the 60° sector of radius 9: A = (60/360)·π·81 = 81π/6 = 13.5π ≈ 42.41 square units; in radians, 60° = π/3 and ½·81·(π/3) = 27π/2 = 13.5π, the same. For the sector of radius 5 with angle 2.4 radians: A = ½·25·2.4 = 30 square units.

A worked model inside the lesson. A sector of a circle of radius 12 has area 24π. What is its central angle, and how long is its arc? The full circle has area 144π, so the sector is 24π/144π = 1/6 of the circle. Its central angle is (1/6)·360° = 60°, which is (1/6)·2π = π/3 radians ≈ 1.047. Its arc length is rθ = 12·(π/3) = 4π ≈ 12.57 units; by the degree formula, (60/360)·2π·12 = 4π as well. Check the area with the radian formula: ½·144·(π/3) = 24π.

Exact and approximate forms. Answers involving π should be given exactly (3π, 13.5π, 5π/6) and then, if a decimal is wanted, approximated with the precision stated (3π ≈ 9.42 to two decimal places). A decimal alone hides whether the computation was exact, and rounding early can make later steps drift.

The misconception: radians are "small degrees." Students sometimes treat a radian as another unit to be converted by a memorized factor without knowing where it comes from, and then confuse which factor to use. Return to the definition: θ = s/r. If the arc is as long as the radius, the angle is 1 radian; since the circumference is a little more than 6 radii, a full turn is a little more than 6 radians, namely 2π ≈ 6.28. That picture also tells you which conversion is right: 1 radian must be more than 57° because 6.28 of them make 360°. A second misconception is using s = rθ with θ in degrees: with r = 5 and θ = 60° a student writes s = 300. The correct arc is 5·(π/3) ≈ 5.24; the formula s = rθ requires radians, because it came from the definition of radian measure.

Worked arc length by similarity: A central angle of 72° intercepts an arc of length 4π on a circle of radius 10. What arc does the same angle intercept on a concentric circle of radius 25? The dilation with scale factor 25/10 = 2.5 maps the small arc onto the large one, so the large arc is 2.5·4π = 10π ≈ 31.42. Check directly: (72/360)·2π·25 = (1/5)·50π = 10π. The ratio s/r is 4π/10 = 2π/5 for the small circle and 10π/25 = 2π/5 for the large one, so the angle is 2π/5 radians ≈ 1.257, and 72·π/180 = 2π/5 agrees.

Worked radian conversion both ways: Convert 225° to radians: 225·π/180 = 5π/4 ≈ 3.927 radians. Convert 7π/6 radians to degrees: (7π/6)(180/π) = 210°. Convert 3 radians to degrees: 3·180/π ≈ 171.89°. Each uses 180° = π radians; in the first, the factor π/180 carries degrees to radians, and in the others 180/π carries radians to degrees.

Worked sector area, derived and applied: A circular garden bed of radius 6 m has a wedge-shaped section with central angle 135°. The wedge is 135/360 = 3/8 of the circle, so its area is (3/8)·π·36 = 13.5π ≈ 42.41 m². In radians, 135° = 3π/4, and ½·36·(3π/4) = 27π/2 = 13.5π m², the same. The arc along the outside of the wedge has length 6·(3π/4) = 9π/2 ≈ 14.14 m, so a border around the whole wedge, two radii plus the arc, is 12 + 9π/2 ≈ 26.14 m.

Worked reverse problem: An arc of length 15 cm subtends a central angle of 2.5 radians. The radius is r = s/θ = 15/2.5 = 6 cm. The sector's area is ½·6²·2.5 = ½·36·2.5 = 45 cm². In degrees the angle is 2.5·180/π ≈ 143.24°, and the degree formula gives (143.24/360)·π·36 ≈ 45.0 cm², agreeing to rounding.

Formula-derivation record: On one page, derive in your own words and symbols: the arc-length fraction formula s = (θ/360)·2πr; the statement that s/r depends only on the angle, with a sentence naming the dilation that proves it; the definition θ = s/r and the formula s = rθ; the two conversion formulas from 180° = π radians; and the sector area formula both as a fraction of πr² and as ½r²θ. Under each, work one example of your own with r and θ chosen by you, giving exact and rounded answers.

Independent performance: A designer is laying out a fan-shaped stage (a sector) and needs its front edge length, its floor area and its angle in both units. Choose a radius between 8 and 20 m and a central angle between 70° and 160°, then compute the arc length and area exactly (in terms of π) and to two decimal places, convert the angle to radians, and recompute both quantities using the radian formulas. Success criteria: the two routes agree; the exact forms are simplified; units are stated (m for the arc, m² for the area, "rad" or no unit for radians); the write-up explains why s = rθ requires the angle in radians and gives the wrong answer that would result from using degrees.