Geometry Arc Length and Sector Area Worksheet Solutions (PDF)

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

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

Use this answer key to check the 20 problems on arc length and sectors. Every solution shows the formula or theorem used, each step of the work, and the final answer in bold. If an answer differs from yours, compare the steps line by line to find where the work changed.

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

Arcs sectors answers: exercises 1 to 10

Exercise 1 – Arc length 40 degrees

\(s = \frac{40}{360} \cdot 2\pi \cdot 9 = 2\pi\).

2π, about 6.28

Exercise 2 – Arc length 210 degrees

\(s = \frac{210}{360} \cdot 2\pi \cdot 9 = \frac{21\pi}{2}\).

21π/2, about 32.99

Exercise 3 – Pie chart slice

A = 17.82π.

Angle: \(0.22 \cdot 360^{\circ} = 79.2^{\circ}\).

Area: \(0.22 \cdot \pi \cdot 9^{2} = 17.82\pi = 56.0\)

79.2°; 17.82π, about 56.0 square centimeters

Exercise 4 – Sector area 90 degrees

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

\(A = \frac{90}{360} \cdot \pi \cdot 10^{2} = 25\pi\)

25π, about 78.54

Exercise 5 – Find the central angle

\(\text{circumference} = 2\pi \cdot 14 = 28\pi\).

\(\frac{7\pi}{28\pi} = \frac{1}{4}\) of the circle.

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

90°

Exercise 6 – Find the radius

\(\frac{90}{360} \cdot \pi r^{2} = 25\pi\)

\(\frac{1}{4}r^{2} = 25\)

\(r^{2} = 100\), so \(r = 10\).

r = 10

Exercise 7 – Degrees to radians

\(60 \cdot \frac{\pi}{180} = \frac{\pi}{3}\)

\(135 \cdot \frac{\pi}{180} = \frac{3\pi}{4}\)

π/3 and 3π/4

Exercise 8 – Radians to degrees (decimal)

\(2.5 \cdot \frac{180}{\pi} = 143.2\)

143.2°

Exercise 9 – Arc length in radians

\(s = 8 \cdot \frac{\pi}{4} = 2\pi\), about 6.28.

\(s = 10 \cdot 1.5 = 15\).

2π (about 6.28) and 15

Exercise 10 – Sector area in radians

\(A = \frac{1}{2}r^{2}\text{theta} = \frac{1}{2} \cdot 25 \cdot \frac{3\pi}{4} = \frac{75\pi}{8}\)

75π/8, about 29.45

Exercises 11 and up

Exercise 11 – Radian measure

\(\text{theta} = \frac{s}{r} = \frac{12}{8} = 1.5\)

1.5 radians

Exercise 12 – Perimeter of a sector

Perimeter = 10 + 10 + arc.

Arc: \(\frac{72}{360} \cdot 2\pi \cdot 10 = 4\pi\).

\(20 + 4\pi\) = 32.57.

20 + 4π, about 32.57

Exercise 13 – Area of a segment

Sector: \(\frac{90}{360} \cdot \pi \cdot 64 = 16\pi\).

Right triangle with legs 8 and 8: \(\frac{1}{2} \cdot 8 \cdot 8 = 32\).

\(16\pi – 32\) = 18.27.

16π − 32, about 18.27

Exercise 14 – Full circle from a sector

A \(60^{\circ}\) sector is \(\frac{1}{6}\) of the circle, so the circle has area \(6 \cdot 24\pi = 144\pi\).

\(r^{2} = 144\), so \(r = 12\).

144π; r = 12

Exercise 15 – Same arc length, different circles

Arc in A: \(\frac{54}{360} \cdot 2\pi \cdot 10 = 3\pi\).

\(\frac{\text{theta}}{360} \cdot 2\pi \cdot 15 = 3\pi\), so \(\text{theta} = \frac{3 \cdot 360}{30} = 36^{\circ}\).

36°

Exercise 16 – Pizza

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

Radius 8. Each slice has central angle \(360^{\circ} \div 8 = 45^{\circ}\).

Crust: \(\frac{45}{360} \cdot 2\pi \cdot 8 = 2\pi\) = 6.28 in.

Area: \(\frac{45}{360} \cdot \pi \cdot 8^{2} = 8\pi\) = 25.13 sq in.

Crust 6.28 in; area 25.13 sq in

Exercise 17 – Garden sprinkler

Watered region.

\(\frac{75}{360} \cdot \pi \cdot 40^{2} = \frac{1000}{3}\pi\)

About 1047 square feet

Exercise 18 – Minute hand

20 minutes is \(\frac{20}{60} = \frac{1}{3}\) of a circle (120°).

\(s = \frac{1}{3} \cdot 2\pi \cdot 7 = \frac{14\pi}{3} = 14.66\)

14.66 inches

Exercise 19 – Track curve

A semicircle is half of the circumference: \(\frac{1}{2} \cdot 2\pi \cdot 36.5 = 36.5\pi\).

114.7 meters

Exercise 20 – Windshield wiper

Area = large sector minus small sector.

\(\frac{100}{360} \cdot \pi (24^{2} – 6^{2}) = \frac{100}{360} \cdot 540\pi = 150\pi\)

About 471 square inches

Key facts behind this arcs sectors answers

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 answers

Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.

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