Arcs and Sectors
A free Geometry lesson from the “Measurement, Circles, and 3D Solids” unit, with a worked example and practice problems including step-by-step solutions.
An arc is part of a circle's circumference, and a sector is part of a circle's area. The central angle tells what fraction of the circle is being used. In Measurement, Circles, and 3D Solids, students need to read the diagram, name the relationship, choose a theorem or formula, and justify why the result follows. The expanded practice now includes fluency, transfer, cumulative review, and proof-style reasoning so Geometry feels connected instead of isolated by topic.
What you'll learn
- Use central angles
- Find arc length
- Find sector area
Worked example
Problem. A sector has central angle 90 degrees. What fraction of the circle is it?
- A full circle is 360 degrees.
- 90/360 = 1/4.
- The sector is one fourth of the circle.
- Connect the result back to Arcs and Sectors so the geometric relationship is explicit.
Answer: 1/4
Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.
Practice problems
1. A central angle is 180 degrees. What fraction of the circle is it?
Show solution
- 180/360 = 1/2.
Answer: 1/2
2. A central angle is 90 degrees. What fraction of the circle is it?
Show solution
- 90/360 = 1/4.
Answer: 1/4
3. A circle has circumference 24. A 90-degree arc has length what?
Show solution
- 1/4 of 24 is 6.
Answer: 6
4. A circle has circumference 30. A 120-degree arc has length what?
Show solution
- 120/360 = 1/3.
- 1/3 of 30 is 10.
Answer: 10
5. A circle has area 144. A 60-degree sector has area what?
Show solution
- 60/360 = 1/6.
- 144/6 = 24.
Answer: 24
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