These 25 problems practice standard G-C.B.5: finding arc lengths and sector areas as fractions of a circle, working with radian measure, and converting between degrees and radians. They move from simple fractions of a circle to word problems about pizza, sprinklers and Ferris wheels.
Show the fraction of the circle first, keep answers exact in terms of \(\pi\), and round only when the problem says so.
Problem 1: Fraction of a circle
A central angle measures \(150^{\circ}\). Write the fraction of the circumference that its arc represents, in lowest terms.
Problem 2: Quarter-circle arc
A circle has radius \(14\text{ yd}\). Find the length of the arc cut off by a central angle of \(90^{\circ}\). Give the exact length and a decimal rounded to the nearest hundredth.
Problem 3: Arc length of a sector
Find the arc length of the sector shown. Give the exact value and a decimal rounded to the nearest hundredth.
Problem 4: Arc with an obtuse angle
A circle has radius \(10\text{ in}\). Find the length of the arc intercepted by a central angle of \(135^{\circ}\). Give the exact length and a decimal rounded to the nearest hundredth.
Problem 5: Area of a sector
Find the area of the shaded sector. Give the exact area and a decimal rounded to the nearest hundredth.
Problem 6: Area with an obtuse angle
Find the area of a sector with radius \(10\text{ in}\) and central angle \(135^{\circ}\). Give the exact area and a decimal rounded to the nearest hundredth.
Problem 7: Central angle from arc length
The arc shown has length \(4\pi\text{ in}\) on a circle of radius \(8\text{ in}\). Find the central angle in degrees.
Problem 8: Radius from arc length
An arc with central angle \(72^{\circ}\) has length \(6\pi\text{ cm}\). Find the radius of the circle.
Problem 9: Radius from sector area
A sector with central angle \(120^{\circ}\) has area \(12\pi\text{ ft}^{2}\). Find the radius.
Problem 10: Central angle from sector area
A sector of a circle with radius \(5\text{ in}\) has area \(5\pi\text{ in}^{2}\). Find the central angle in degrees.
Problem 11: Meaning of a radian
On a circle with radius \(5\text{ cm}\), a central angle intercepts an arc of length \(15\text{ cm}\). Find the measure of the angle in radians.
Problem 12: Radians from arc and radius
A circle has radius \(4\text{ in}\). A central angle intercepts an arc \(10\text{ in}\) long. Find the angle in radians.
Problem 13: Degrees to radians
Convert \(150^{\circ}\) to radians. Give the answer in terms of \(\pi\).
Problem 14: Reflex angle in radians
Convert \(225^{\circ}\) to radians. Give the answer in terms of \(\pi\).
Problem 15: Radians to degrees
Convert \(\frac{7\pi}{6}\) radians to degrees.
Problem 16: Radians to degrees, again
Convert \(\frac{5\pi}{3}\) radians to degrees.
Problem 17: A radian without pi
Convert 2 radians to degrees. Round to the nearest hundredth of a degree.
Problem 18: Arc length with radians
A central angle of \(\frac{\pi}{3}\) radians is drawn in a circle of radius \(7\text{ in}\). Find the arc length. Give the exact value and a decimal rounded to the nearest hundredth.
Problem 19: Sector area with radians
A sector has radius \(6\text{ ft}\) and central angle \(\frac{\pi}{4}\) radians. Find its area. Give the exact value and a decimal rounded to the nearest hundredth.
Problem 20: Pizza slice
Jamal cuts a 16-inch pizza into 8 equal slices. Find (a) the length of the crust on one slice and (b) the area of one slice. Give decimals rounded to the nearest hundredth.
Problem 21: Lawn sprinkler
A sprinkler at the edge of a lawn sprays water up to 25 ft away and turns back and forth through an angle of \(80^{\circ}\). Find the area of grass it waters. Round to the nearest square foot.
Problem 22: Running the curve
Maya runs along a circular track of radius 30 yd. She starts at one point and stops after covering a central angle of \(200^{\circ}\). How far does she run? Round to the nearest tenth of a yard.
Problem 23: Clock hand
A minute hand is 6 in long. In 20 minutes it turns through \(120^{\circ}\). How far does its tip travel? Give the exact distance and a decimal rounded to the nearest hundredth.
Problem 24: Pie chart slice
In a pie chart of radius 5 in, the slice for the school store covers 30% of the circle. Find (a) the central angle of the slice, (b) the length of its curved edge and (c) its area. Give decimals rounded to the nearest hundredth.
Problem 25: Ferris wheel
A Ferris wheel at a county fair has radius 40 ft. A rider travels 100 ft along the circle. Find the angle the rider turns through (a) in radians and (b) in degrees, rounded to the nearest hundredth.
See every chapter on the Geometry math page.