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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.

What you'll learn

Why it matters: Pizza slices, clock faces, gears, and sports-field markings all use sectors and arcs to describe part of a circle.

Worked example

Problem. A sector has central angle 90 degrees. What fraction of the circle is it?

  1. A full circle is 360 degrees.
  2. 90/360 = 1/4.
  3. The sector is one fourth of the circle.

Answer: 1/4

Practice problems

1. A central angle is 180 degrees. What fraction of the circle is it?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A central angle is 180 degrees. What fraction of the circle is it?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 180/360 = 1/2.
  4. Check the result by substituting or estimating: the response should match 1/2 and make sense in the original problem.

Answer: 1/2

2. A central angle is 90 degrees. What fraction of the circle is it?

Show solution
  1. Warm-up: First identify exactly what the question is asking: A central angle is 90 degrees. What fraction of the circle is it?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 90/360 = 1/4.
  4. Check the result by substituting or estimating: the response should match 1/4 and make sense in the original problem.

Answer: 1/4

3. A circle has circumference 24. A 90-degree arc has length what?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A circle has circumference 24. A 90-degree arc has length what?
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. 1/4 of 24 is 6.
  4. Check the result by substituting or estimating: the response should match 6 and make sense in the original problem.

Answer: 6

4. A circle has circumference 30. A 120-degree arc has length what?

Show solution
  1. Core Practice: First identify exactly what the question is asking: A circle has circumference 30. A 120-degree arc has length what?
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. 120/360 = 1/3.
  4. 1/3 of 30 is 10.
  5. Check the result by substituting or estimating: the response should match 10 and make sense in the original problem.

Answer: 10

5. A circle has area 144. A 60-degree sector has area what?

Show solution
  1. Challenge: First identify exactly what the question is asking: A circle has area 144. A 60-degree sector has area what?
  2. For circle problems, connect the formula or theorem to the given radius, diameter, chord, arc, or center information.
  3. 60/360 = 1/6.
  4. 144/6 = 24.
  5. Check the result by substituting or estimating: the response should match 24 and make sense in the original problem.

Answer: 24

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