Geometry Arc Length and Sector Area Worksheet (PDF)

Geometry math worksheet: Arc Length and Sector Area – Maths PDF

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This arcs sectors worksheet for Geometry covers arc length and sector area (Common Core G-C.B.5), with a free PDF to print.

These 20 problems practice arc length, sector area and radian measure (Common Core G-C.B.5). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are applications with pizzas, sprinklers, clocks, tracks and wipers.

Geometry · Common Core State Standards: G-C.B.5

Arcs sectors worksheet: exercises 1 to 10

Exercise 1 – Arc length 40 degrees

Find the length of an arc with central angle \(40^{\circ}\) in a circle of radius 9. Give an exact answer and a decimal rounded to hundredths.

Exercise 2 – Arc length 210 degrees

Find the length of an arc with central angle \(210^{\circ}\) in a circle of radius 9. Give the exact value and a decimal.

Exercise 3 – Pie chart slice

A slice for 22% of the chart.

A pie chart has radius 9 cm. The slice labeled Rent is 22% of the whole chart. Find its central angle and its area.

Exercise 4 – Sector area 90 degrees

Sector area = (90/360) of the circle’s area = 25π.

Find the area of a sector with central angle \(90^{\circ}\) and radius 10.

Exercise 5 – Find the central angle

An arc of length \(7\pi\) lies on a circle with radius 14. Find the central angle in degrees.

Exercise 6 – Find the radius

A sector with central angle \(90^{\circ}\) has area \(25\pi\). Find the radius of the circle.

Exercise 7 – Degrees to radians

Convert to radians: \(60^{\circ}\) and \(135^{\circ}\).

Exercise 8 – Radians to degrees (decimal)

A turntable rotates through 2.5 radians. Express this turn in degrees to the nearest tenth.

Exercise 9 – Arc length in radians

One radian: the arc has the same length as the radius.

Use \(s = r\text{theta}\) to find the arc length for \(r = 8\) and \(\text{theta} = \frac{\pi}{4}\). Then find the arc length for \(r = 10\) and \(\text{theta} = 1.5\) radians.

Exercise 10 – Sector area in radians

Find the area of a sector with \(r = 5\) and \(\text{theta} = \frac{3\pi}{4}\) radians.

Exercises 11 and up

Exercise 11 – Radian measure

An arc 12 inches long is on a circle of radius 8 inches. Find the central angle in radians.

Exercise 12 – Perimeter of a sector

A sector has radius 10 and central angle \(72^{\circ}\). Find its perimeter (two radii plus the arc).

Exercise 13 – Area of a segment

Sector with a right angle at the center.

A circular segment is bounded by a chord and its arc. Find the area of the segment cut off by a chord that subtends \(90^{\circ}\) in a circle of radius 8. (Subtract the area of the right triangle from the sector.)

Exercise 14 – Full circle from a sector

A sector of \(60^{\circ}\) has area \(24\pi\). Find the area of the whole circle and its radius.

Exercise 15 – Same arc length, different circles

Circle A has radius 10 and an arc of \(54^{\circ}\). Circle B has radius 15. What central angle in circle B gives an arc of the same length?

Exercise 16 – Pizza

A pizza cut into 8 equal slices: each slice is a sector of 45°.

A 16-inch pizza (diameter 16) is cut into 8 equal slices by Sofia. Find the length of the crust on one slice and the area of one slice.

Exercise 17 – Garden sprinkler

A garden sprinkler waters a sector of \(75^{\circ}\) with a reach of 40 feet. Find the area watered, to the nearest square foot.

Exercise 18 – Minute hand

A minute hand is 7 inches long. How far does its tip travel in 20 minutes? Round to the nearest hundredth.

Exercise 19 – Track curve

The inside lane of a running track has a semicircular curve of radius 36.5 meters at each end. Find the length of one curve to the nearest tenth.

Exercise 20 – Windshield wiper

A windshield wiper sweeps through \(100^{\circ}\). Its blade covers the region from 6 inches to 24 inches from the pivot. Find the area cleaned to the nearest square inch.

Key facts behind this arcs sectors worksheet

Key facts

Arc length = (angle/360) x 2 pi r.
Sector area = (angle/360) x pi r squared.
180 degrees = pi radians.
In radians: arc length s = r times theta, area A = (1/2) r squared theta.
A radian is the central angle whose arc length equals the radius.

How to use this arcs sectors worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the arcs sectors worksheet as a PDF with the button below and keep it with your notes.

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