Arc Length and Sector Area

Arc Length and Sector Area

Learn how to find the arc length and sector area of a circle using the following step-by-step guide with examples.

Step by step guide to find arc length and sector area of circles

  • To find a sector of a circle, use this formula: Area of a sector \(=\color{blue}{πr^2 (\frac{θ}{360})}\)
    \(r\) is the radius of the circle and θ is the central angle of the sector.
  • To find the arc of a sector of a circle, use this formula: Arc of a sector \(=\color{blue}{(\frac{θ}{180})πr}\)

Example 1:

Find the length of the arc. Round your answers to the nearest tenth. \((π=3.14) r=24\) cm, \(\theta=60^\circ\)

Solution:

Use this formula: length of a sector \(=\color{blue}{(\frac{θ}{180})πr}\)
length of a sector \(=(\frac{60}{180})π(24)=(\frac{1}{3})π(24)=8×3.14 \cong 25.12 \) cm

Example 2:

Find the area of the sector. \(r=6\) ft, \(\theta=90^\circ\)

Solution:

Use this formula: Area of a sector \(=\color{blue}{πr^2 (\frac{θ}{360})}\)
Area of a sector \(=πr^2 (\frac{θ}{360})=(3.14)(6^2 )(\frac{90}{360})=(3.14)(36)(\frac{1}{4})≅28.26 \)

Example 3:

Find the length of the arc. Round your answers to the nearest tenth. \((π=3.14) r=28\) cm, \(\theta=45^\circ\)

Solution:

Use this formula: length of a sector \(=\color{blue}{(\frac{θ}{180})πr}\)
length of a sector \(=(\frac{45}{180})π(28)=(\frac{1}{4})π(28)=7×3.14 \cong 22\) cm

Example 4:

Find the area of the sector. \(r=5 \) ft, \(\theta=80^\circ\)

Solution:

Use this formula: Area of a sector \(=\color{blue}{πr^2 (\frac{θ}{360})}\)
Area of a sector \(=πr^2 (\frac{θ}{360})=(3.14)(5^2 )(\frac{80}{360})≅17.4 \)

Exercises

Find the length of each arc. Round your answers to the nearest tenth.

  • \(\color{blue}{r = 28 \ cm, \theta = 45^\circ}\)
  • \(\color{blue}{r = 15 \ ft, \theta = 95^\circ}\)
  • \(\color{blue}{r = 22 \ ft, \theta = 60^\circ}\)
  • \(\color{blue}{r = 12 \ m, \theta = 85^\circ}\)

Download Arc Length and Sector Area Worksheet

  • \(\color{blue}{22 \ cm}\)
  • \(\color{blue}{25 \ ft}\)
  • \(\color{blue}{23 \ ft}\)
  • \(\color{blue}{18 \ m}\)

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