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

Geometry Arc Length Worksheet Answer Key: problems 1–5

Problem 1: Fraction of a circle

Fraction = angle \(: 360^{\circ}\): \(\frac{150}{360}=\frac{5}{12}\).

\(\frac{5}{12}\) of the circumference

Problem 2: Quarter-circle arc

Fraction of the circle: \(\frac{90}{360}=\frac{1}{4}\).

Circumference: \(2\pi r=2\pi(14)=28\pi\text{ yd}\).

Arc length: \(\frac{1}{4}\cdot28\pi=7\pi\text{ yd}\approx 21.99\text{ yd}\).

Arc length = \(7\pi\text{ yd}\approx 21.99\text{ yd}\)

Problem 3: Arc length of a sector

Fraction of the circle: \(\frac{75}{360}=\frac{5}{24}\).

Circumference: \(2\pi r=2\pi(12)=24\pi\text{ ft}\).

Arc length: \(\frac{5}{24}\cdot24\pi=5\pi\text{ ft}\approx 15.71\text{ ft}\).

Arc length = \(5\pi\text{ ft}\approx 15.71\text{ ft}\)

Problem 4: Arc with an obtuse angle

Fraction of the circle: \(\frac{135}{360}=\frac{3}{8}\).

Circumference: \(2\pi r=2\pi(10)=20\pi\text{ in}\).

Arc length: \(\frac{3}{8}\cdot20\pi=\frac{15\pi}{2}\text{ in}\approx 23.56\text{ in}\).

Arc length = \(\frac{15\pi}{2}\text{ in}\approx 23.56\text{ in}\)

Problem 5: Area of a sector

Fraction of the circle: \(\frac{60}{360}=\frac{1}{6}\).

Area of the whole circle: \(\pi r^{2}=\pi(6)^{2}=36\pi\text{ in}^{2}\).

Sector area: \(\frac{1}{6}\cdot36\pi=6\pi\text{ in}^{2}\approx 18.85\text{ in}^{2}\).

Sector area = \(6\pi\text{ in}^{2}\approx 18.85\text{ in}^{2}\)

Geometry Arc Length Worksheet Answer Key: problems 6–10

Problem 6: Area with an obtuse angle

Fraction of the circle: \(\frac{135}{360}=\frac{3}{8}\).

Area of the whole circle: \(\pi r^{2}=\pi(10)^{2}=100\pi\text{ in}^{2}\).

Sector area: \(\frac{3}{8}\cdot100\pi=\frac{75\pi}{2}\text{ in}^{2}\approx 117.81\text{ in}^{2}\).

Sector area = \(\frac{75\pi}{2}\text{ in}^{2}\approx 117.81\text{ in}^{2}\)

Problem 7: Central angle from arc length

Circumference: \(2\pi(8)=16\pi\text{ in}\).

Fraction of the circle: \(\frac{4\pi}{16\pi}=\frac{1}{4}\).

Angle: \(\frac{1}{4}\cdot 360^{\circ}=90^{\circ}\).

\(\theta=90^{\circ}\)

Problem 8: Radius from arc length

Fraction: \(\frac{72}{360}=\frac{1}{5}\). So \(\frac{1}{5}\cdot 2\pi r=6\pi\).

Multiply both sides by 5: \(2\pi r=30\pi\). Divide by \(2\pi\): \(r=15\).

Check: \(\frac{1}{5}\cdot 2\pi(15)=6\pi\). ✓

\(r=15\text{ cm}\)

Problem 9: Radius from sector area

Fraction: \(\frac{120}{360}=\frac{1}{3}\). So \(\frac{1}{3}\pi r^{2}=12\pi\).

Multiply by 3: \(\pi r^{2}=36\pi\). Divide by \(\pi\): \(r^{2}=36\), so \(r=6\).

Check: \(\frac{1}{3}\pi(6)^{2}=12\pi\). ✓

\(r=6\text{ ft}\)

Problem 10: Central angle from sector area

Area of the whole circle: \(\pi(5)^{2}=25\pi\text{ in}^{2}\).

Fraction: \(\frac{5\pi}{25\pi}=\frac{1}{5}\).

Angle: \(\frac{1}{5}\cdot 360^{\circ}=72^{\circ}\).

\(72^{\circ}\)

Geometry Arc Length Worksheet Answer Key: problems 11–15

Problem 11: Meaning of a radian

Radian measure = arc length \(: \) radius: \(\theta=\frac{s}{r}=\frac{15}{5}=3\).

3 radians

Problem 12: Radians from arc and radius

\(\theta=\frac{s}{r}=\frac{10}{4}=\frac{5}{2}=2.5\).

\(\frac{5}{2}\) radians (2.5 radians)

Problem 13: Degrees to radians

Multiply by \(\frac{\pi}{180}\): \(150\cdot\frac{\pi}{180}=\frac{150\pi}{180}=\frac{5\pi}{6}\).

\(\frac{5\pi}{6}\) radians

Problem 14: Reflex angle in radians

\(225\cdot\frac{\pi}{180}=\frac{225\pi}{180}=\frac{5\pi}{4}\) (divide top and bottom by 45).

\(\frac{5\pi}{4}\) radians

Problem 15: Radians to degrees

Multiply by \(\frac{180}{\pi}\): \(\frac{7\pi}{6}\cdot\frac{180}{\pi}=\frac{7\cdot 180}{6}=7\cdot 30=210^{\circ}\).

\(210^{\circ}\)

Geometry Arc Length Worksheet Answer Key: problems 16–20

Problem 16: Radians to degrees, again

\(\frac{5\pi}{3}\cdot\frac{180}{\pi}=5\cdot 60=300^{\circ}\).

\(300^{\circ}\)

Problem 17: A radian without pi

\(2\cdot\frac{180}{\pi}=\frac{360}{\pi}\approx 114.59^{\circ}\).

\(\approx 114.59^{\circ}\)

Problem 18: Arc length with radians

With radians, \(s=r\theta\): \(s=7\cdot\frac{\pi}{3}=\frac{7\pi}{3}\approx 7.33\text{ in}\).

\(\frac{7\pi}{3}\text{ in}\approx 7.33\text{ in}\)

Problem 19: Sector area with radians

With radians, \(A=\frac{1}{2}r^{2}\theta\): \(A=\frac{1}{2}(6)^{2}\cdot\frac{\pi}{4}=18\cdot\frac{\pi}{4}=\frac{9\pi}{2}\approx 14.14\text{ ft}^{2}\).

Check with degrees: \(\frac{\pi}{4}=45^{\circ}\), and \(\frac{45}{360}\cdot 36\pi=\frac{9\pi}{2}\). ✓

\(\frac{9\pi}{2}\text{ ft}^{2}\approx 14.14\text{ ft}^{2}\)

Problem 20: Pizza slice

The diameter is 16 in, so the radius is 8 in. Each slice has central angle \(360^{\circ}: 8=45^{\circ}\), which is \(\frac{1}{8}\) of the pizza.

(a) Crust: \(\frac{1}{8}\cdot 2\pi(8)=\frac{1}{8}\cdot 16\pi=2\pi\approx 6.28\text{ in}\).

(b) Area: \(\frac{1}{8}\cdot\pi(8)^{2}=\frac{1}{8}\cdot 64\pi=8\pi\approx 25.13\text{ in}^{2}\).

(a) \(2\pi\approx 6.28\text{ in}\); (b) \(8\pi\approx 25.13\text{ in}^{2}\)

Geometry Arc Length Worksheet Answer Key: problems 21–25

Problem 21: Lawn sprinkler

Fraction of the circle: \(\frac{80}{360}=\frac{2}{9}\).

Area of the whole circle: \(\pi r^{2}=\pi(25)^{2}=625\pi\text{ ft}^{2}\).

Sector area: \(\frac{2}{9}\cdot625\pi=\frac{1250\pi}{9}\text{ ft}^{2}\approx 436.33\text{ ft}^{2}\).

Sector area = \(\frac{1250\pi}{9}\text{ ft}^{2}\approx 436.33\text{ ft}^{2}\)

Rounded: about 436 square feet.

Problem 22: Running the curve

Fraction of the circle: \(\frac{200}{360}=\frac{5}{9}\).

Circumference: \(2\pi r=2\pi(30)=60\pi\text{ yd}\).

Arc length: \(\frac{5}{9}\cdot60\pi=\frac{100\pi}{3}\text{ yd}\approx 104.72\text{ yd}\).

Arc length = \(\frac{100\pi}{3}\text{ yd}\approx 104.72\text{ yd}\)

Rounded to the nearest tenth: 104.7 yd.

Problem 23: Clock hand

The tip moves along an arc of radius 6 in. Fraction: \(\frac{120}{360}=\frac{1}{3}\) (also \(\frac{20}{60}\) of an hour).

Arc length: \(\frac{1}{3}\cdot 2\pi(6)=\frac{1}{3}\cdot 12\pi=4\pi\approx 12.57\text{ in}\).

\(4\pi\approx 12.57\text{ in}\)

Problem 24: Pie chart slice

(a) \(0.30\cdot 360^{\circ}=108^{\circ}\).

(b) Arc: \(0.30\cdot 2\pi(5)=0.30\cdot 10\pi=3\pi\approx 9.42\text{ in}\).

(c) Area: \(0.30\cdot\pi(5)^{2}=0.30\cdot 25\pi=\frac{15\pi}{2}\approx 23.56\text{ in}^{2}\).

(a) \(108^{\circ}\); (b) \(3\pi\approx 9.42\text{ in}\); (c) \(\frac{15\pi}{2}\approx 23.56\text{ in}^{2}\)

Problem 25: Ferris wheel

(a) \(\theta=\frac{s}{r}=\frac{100}{40}=2.5\) radians.

(b) \(2.5\cdot\frac{180}{\pi}=\frac{450}{\pi}\approx 143.24^{\circ}\).

(a) 2.5 radians; (b) \(\approx 143.24^{\circ}\)

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