Derive arc scaling and sector area

Try

A circle of radius 6 cm contains a sector whose central angle is 120 degrees. It occupies one third of a full turn. Predict its arc length and area by comparing it with the whole circle, rather than selecting a formula from memory.

The circumference is 12π cm and the circle area is 36π cm², so the arc is 4π cm and the sector area is 12π cm². Here arc length measures the curved boundary, while sector area measures the region between two radii and their chosen arc. They have different units. The two radii also belong to the sector boundary; its perimeter is not just the arc length.

Predict

Enlarge the entire figure by a positive factor k. Predict whether the central angle changes. It does not: a dilation preserves angles. Straight segment lengths multiply by k, so a polygonal chain approximating the arc has k times its original length. As those chains approximate the curve more closely, the arc length has the same scale factor. Areas multiply by k², as can be seen by subdividing an inscribed polygonal sector into triangles.

Consequently the ratio arc length divided by radius does not change between similar circles. This scale-independent ratio is the radian measure θ of the selected central angle: θ=s/r, where s is arc length and r>0 is radius. For one full turn, θ=2πr/r=2π.

Experiment

Construct the radius-6 example, marking the chosen 120-degree arc explicitly. There is also a 240-degree arc between the same endpoints; naming endpoints alone would not choose one.

Make a second copy with radius 3 but the same angle. Its arc is 2π and its area is 3π. Confirm that s/r remains 2π/3 in both copies while area/r² remains π/3. The two ratios describe different scaling laws.

Understand

Why may an angle fraction be used as a length or area fraction? Divide a circle into n congruent sectors using equal central angles. Rotations carry one sector onto the next, preserving arc lengths and areas. By additivity, m consecutive sectors have m/n of the circumference and m/n of the disk area. This proves the rule for rational fractions of a turn.

An arbitrary angle can be trapped between rational fractions approaching it from below and above. The corresponding sectors contain and are contained in the target sector. Their areas and arc lengths bound the target values, with gaps that approach zero. This uses ordinary monotonicity and continuity of geometric length and area; it does not pretend that every angle is a rational number of degrees. The same inscribed-polygon limiting interpretation supports the earlier arc-length scaling argument.

For 0≤θ≤2π measured in radians, the turn fraction is θ/(2π). Therefore

s=(θ/(2π))(2πr)=rθ, and A=(θ/(2π))(πr²)=r²θ/2.

The numerical example has θ=2π/3, giving s=6·2π/3=4π and A=36·(2π/3)/2=12π. If θ is supplied in degrees, first convert by multiplying by π/180. Inserting 120 directly into rθ would silently treat degrees as radians.

A signed angle describes direction of rotation. Physical length and ordinary region area are nonnegative. Likewise, a sweep longer than a full turn revisits points; its accumulated travel length differs from the length of a single simple arc, and repeatedly swept area differs from the union of the covered region.

Apply

For a fresh sector, state the radius, chosen arc and angle unit before calculating. Give arc length and area. For a proper sector with 0 < θ < 2π, its full perimeter is s+2r. At θ=2π the region is the whole disk and its boundary is only the circumference, 2πr; the radii are internal. The zero-angle case is degenerate and needs a separately specified boundary convention. Check units at every stage. If the question gives an arc and radius, compute θ=s/r and explain why the ratio is the same on any scaled copy.

Repair the claim that doubling radius while keeping angle fixed doubles sector area. The two radii scale every triangular piece in two directions, so the area quadruples. Doubling arc length and quadrupling area are consistent, not contradictory.

Continue

The radian definition connects geometry to trigonometric graphs and, later, calculus. No integration is needed for these sector formulas. The circle circumference and area results are established earlier through geometric limits; this lesson uses them rather than deriving them from the sector formula in a circle.

Optional depth: distinguish a circular segment, cut off by a chord, from a sector bounded by two radii. Its area can require subtracting a triangle from the appropriate sector, with attention to whether the chosen arc is minor or major.