This geometry arc length test answer key gives the full solution and the scoring guide for each of the 9 test questions on arc length, sector area and radians.
Use this key to score the 20-point test. Each answer shows full work, and the scoring note after it says how the points are awarded. Give partial credit for a correct setup even when the arithmetic has a slip, and take away no points for leaving the answer exact instead of rounded when the question allows it.
Geometry Arc Length Test Answer Key: questions 1–5
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)
- Circumference \(24\pi\); fraction \(\frac{8\pi}{24\pi}=\frac{1}{3}\); angle \(\frac{1}{3}\cdot 360^{\circ}=120^{\circ}\).
- 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)
- \(135\cdot\frac{\pi}{180}=\frac{3\pi}{4}\)
- \(300\cdot\frac{\pi}{180}=\frac{5\pi}{3}\)
- \(20\cdot\frac{\pi}{180}=\frac{\pi}{9}\)
Scoring: 2 points if all three are correct, 1 point if two are correct, 0 otherwise.
Geometry Arc Length Test Answer Key: questions 6–9
Question 6: Radians to degrees (2 points)
- \(\frac{11\pi}{6}\cdot\frac{180}{\pi}=330^{\circ}\)
- \(\frac{3\pi}{2}\cdot\frac{180}{\pi}=270^{\circ}\)
- \(\frac{2\pi}{5}\cdot\frac{180}{\pi}=72^{\circ}\)
Scoring: 2 points if all three are correct, 1 point if two are correct, 0 otherwise.
Question 7: Radius from sector area (3 points)
- \(\frac{150}{360}\pi r^{2}=60\pi\), so \(\frac{5}{12}r^{2}=60\) and \(r^{2}=144\), giving \(r=12\text{ ft}\).
- \(\frac{5}{12}\cdot 2\pi(12)=\frac{5}{12}\cdot 24\pi=10\pi\text{ ft}\).
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 spray distance as a radius of 15 ft. Fraction: \(\frac{140}{360}=\frac{7}{18}\).
Area: \(\frac{7}{18}\cdot\pi(15)^{2}=\frac{7}{18}\cdot 225\pi=87.5\pi\approx 274.89\), so about 275 square feet.
Scoring: 1 point for the fraction and the area formula with radius 15, 1 point for 87.5 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.
Scoring summary
20 points: add the points of every question of this geometry arc length test answer key.
How to use this geometry arc length test answer key
Use this geometry arc length test answer key to score the test fairly and to see which skills need more practice.
Printable PDF · Test Answer Key
Geometry Arc Length, Sector Area and Radians test answer key
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