Answer key: Arc Length & Sector Area Worksheet
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Goal: Find arc lengths, sector areas and chords, connect degrees to radians, and see how each measurement changes with the angle and the radius.
Completed results table
| Setting | Arc length | Sector area | Chord length | Fraction of circle |
|---|---|---|---|---|
| r = 5, θ = 30° | 2.618 | 6.545 units² | 2.588 | 8.333% |
| r = 5, θ = 60° | 5.236 | 13.09 units² | 5 | 16.667% |
| r = 5, θ = 90° | 7.854 | 19.635 units² | 7.071 | 25% |
| r = 5, θ = 180° | 15.708 | 39.27 units² | 10 | 50% |
| r = 5, θ = 270° | 23.562 | 58.905 units² | 7.071 | 75% |
| r = 10, θ = 90° | 15.708 | 78.54 units² | 14.142 | 25% |
Answers
Convert 60° to radians, then use s = rθ to find the arc length by hand.
60° × π ÷ 180 = π/3 ≈ 1.047 radians, so s = 5 × π/3 ≈ 5.236.
Compare the 90° sector of radius 5 with the 90° sector of radius 10. What happened to the arc length and the sector area when the radius doubled?
The arc length doubled (7.854 → 15.708), but the area became 4 times as large (19.635 → 78.54). Arc length is proportional to r, while sector area is proportional to r².
Explain why the chord for 60° is exactly 5, the same as the radius.
The chord and the two radii form a triangle with two sides of 5 and a 60° angle between them. It is isosceles, so the other two angles are equal: (180° − 60°) ÷ 2 = 60°. All three angles are 60°, so the triangle is equilateral and the chord is 5.
Why is the chord for 180° equal to 10? Why is the chord for 270° the same as the chord for 90°?
At 180° the chord is a diameter: 2 × 5 = 10. The 270° sector and the 90° sector meet the circle at the same two points, so they share the same chord, about 7.071.
An arc on a circle of radius 5 is 10 units long. Find the central angle in radians and in degrees.
θ = s ÷ r = 10 ÷ 5 = 2 radians, which is 2 × 180 ÷ π ≈ 114.592°.
Show that the degree formula A = (θ ÷ 360) × πr² gives the same sector area as A = ½r²θ for a 90° sector of radius 5.
(90 ÷ 360) × π × 25 = 25π ÷ 4 ≈ 19.635, and ½ × 25 × π/2 = 25π ÷ 4 ≈ 19.635. Both are a quarter of the circle’s area.