Arc Length, Sector Area and Radians: Geometry Math Test Answer Key – PDF Download

Question 1: Arc length (2 points)

Fraction: \(\frac{48}{360}=\frac{2}{15}\). Circumference: \(2\pi(15)=30\pi\).

Arc: \(\frac{2}{15}\cdot 30\pi=4\pi\approx 12.57\text{ in}\).

Scoring: 1 point for the fraction of the circle, 1 point for the exact and rounded arc length with units.

Question 2: Sector area (2 points)

Fraction: \(\frac{200}{360}=\frac{5}{9}\). Whole circle: \(\pi(9)^{2}=81\pi\).

Sector: \(\frac{5}{9}\cdot 81\pi=45\pi\approx 141.37\text{ cm}^{2}\).

Scoring: 1 point for the fraction and the area of the whole circle, 1 point for the exact and rounded sector area with square units.

Question 3: Angle and area from arc (3 points)

1. Circumference \(24\pi\); fraction \(\frac{8\pi}{24\pi}=\frac{1}{3}\); angle \(\frac{1}{3}\cdot 360^{\circ}=120^{\circ}\).
2. Area \(\frac{1}{3}\cdot\pi(12)^{2}=\frac{1}{3}\cdot 144\pi=48\pi\approx 150.80\text{ in}^{2}\).

Scoring: Part (a): 1 point. Part (b): 1 point for using the fraction 1/3 of the area of the whole circle, 1 point for the correct exact and rounded area with units.

Question 4: Radian measure (2 points)

\(\theta=\frac{s}{r}=\frac{21}{7}=3\) radians.

One radian is the angle whose arc is exactly as long as the radius, so this arc is 3 radii long.

Scoring: 1 point for 3 radians, 1 point for a correct explanation: the arc length equals the radius.

Question 5: Degrees to radians (2 points)

1. \(135\cdot\frac{\pi}{180}=\frac{3\pi}{4}\)
2. \(300\cdot\frac{\pi}{180}=\frac{5\pi}{3}\)
3. \(20\cdot\frac{\pi}{180}=\frac{\pi}{9}\)

Scoring: 2 points if all three are correct, 1 point if two are correct, 0 otherwise.

Question 6: Radians to degrees (2 points)

1. \(\frac{11\pi}{6}\cdot\frac{180}{\pi}=330^{\circ}\)
2. \(\frac{3\pi}{2}\cdot\frac{180}{\pi}=270^{\circ}\)
3. \(\frac{2\pi}{5}\cdot\frac{180}{\pi}=72^{\circ}\)

Scoring: 2 points if all three are correct, 1 point if two are correct, 0 otherwise.

Question 7: Radius from sector area (3 points)

1. \(\frac{150}{360}\pi r^{2}=60\pi\), so \(\frac{5}{12}r^{2}=60\) and \(r^{2}=144\), giving \(r=12\text{ ft}\).
2. \(\frac{5}{12}\cdot 2\pi(12)=\frac{5}{12}\cdot 24\pi=10\pi\text{ ft}\).

Scoring: Part (a): 1 point for setting up the equation with the fraction 5/12, 1 point for r = 12 ft. Part (b): 1 point.

Question 8: Windshield wiper (2 points)

Treat the blade as a radius of 24 in. Fraction: \(\frac{110}{360}=\frac{11}{36}\).

Area: \(\frac{11}{36}\cdot\pi(24)^{2}=\frac{11}{36}\cdot 576\pi=176\pi\approx 552.92\), so about 553 square inches.

Scoring: 1 point for the fraction and the area formula with radius 24, 1 point for 176 pi and the rounded value.

Question 9: Arc and area in radians (2 points)

Arc length: \(s=r\theta=10\cdot\frac{2\pi}{5}=4\pi\text{ in}\).

Area: \(A=\frac{1}{2}r^{2}\theta=\frac{1}{2}(100)\cdot\frac{2\pi}{5}=20\pi\text{ in}^{2}\).

Scoring: 1 point for the arc length 4 pi in, 1 point for the area 20 pi square inches.

⬇ Download the Geometry Arc Length, Sector Area and Radians test answer key (PDF, 93 KB)

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