Every solution shows the fraction of the circle, the whole circumference or area, and the final answer in bold with units. Compare each step with your own work, and check that arc lengths are in units of length and areas in square units.
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 = \(6\pi\text{ in}^{2}\approx 18.85\text{ in}^{2}\)
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: \(\frac{1}{5}\cdot 360^{\circ}=72^{\circ}\).
\(72^{\circ}\)
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 \(\frac{180}{\pi}\): \(\frac{7\pi}{6}\cdot\frac{180}{\pi}=\frac{7\cdot 180}{6}=7\cdot 30=210^{\circ}\).
\(210^{\circ}\)
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) \(8\pi\approx 25.13\text{ in}^{2}\)
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) \(\approx 143.24^{\circ}\)
See every chapter on the Geometry math page.