Arc Length and Sector Area Test Answer Key with Scoring – Geometry

Question 1 Arc Length (2 points)

s = 84360 · 2π(7.5) = 730 · 15π = 7π2 ≈ 11.00.

7π/2 in ≈ 11.00 in

Scoring: 1 point for the fraction setup, 1 point for the answer.

Question 2 Sector Area (2 points)

Area = 105360 · π(14²) = 724 · 196π = 343π6 ≈ 179.59.

≈ 179.6 m²

Scoring: 1 point for the setup, 1 point for the answer.

Question 3 Convert the Angles (2 points)

(a) 165 · π180 = 11π12.

(b) 5π9 · 180π = 100°.

(a) 11π/12 (b) 100°

Scoring: 1 point each.

Question 4 Radians from an Arc (3 points)

(a) θ = 9.6 ÷ 4 = 2.4 radians.

(b) 2.4 · 180π ≈ 137.51°.

(c) Area = ½ · 4² · 2.4 = 19.2.

(a) 2.4 rad (b) ≈ 137.5° (c) 19.2 cm²

Scoring: 1 point per part.

Question 5 Radius from Area (3 points)

135360 = 38. So 38πr² = 96π, giving r² = 96 · 83 = 256.

r = 16.

r = 16 ft

Scoring: 1 point for the fraction, 1 point for the equation, 1 point for the radius.

Question 6 Perimeter of a Sector (3 points)

Arc = rθ = 12 · 1.5 = 18. Perimeter = 12 + 12 + 18 = 42.

Arc = 18 cm; perimeter = 42 cm

Scoring: 1 point for the arc length, 1 point for adding two radii, 1 point for the answer.

Question 7 Security Light (3 points)

Area = 140360 · π(30²) = 718 · 900π = 350π ≈ 1099.56.

Far edge = 718 · 60π = 70π3 ≈ 73.3.

Area ≈ 1099.6 ft²; edge ≈ 73.3 ft

Scoring: 1 point for each correct formula, 1 point for both answers.

Question 8 Comparing Sectors (2 points)

Arc lengths scale with the radius: 10 ÷ 4 = 2.5. Areas scale with the square of the radius: 2.5² = 6.25.

Arcs 5 : 2 (2.5); areas 25 : 4 (6.25)

Scoring: 1 point for each ratio.