## Related Topics

- How to Find the Area and Circumference of Circles
- How to Find Equation of a Circle
- How to Find the Center and the Radius of Circles

## Formulas of Arc Length and Sector Area

- To find the area of a sector of a circle, use this formula:

The area of a sector \(=πr^2 (\frac{θ}{360})\), \(r\) 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 \(=(\frac{θ}{180})πr\)

### Arc Length and Sector Area – Example 1:

Find the length of the arc. Round your answers to the nearest tenth. \((π=3.14)\) , \(r=20 cm\) , \(\theta=30^\circ\)

**Solution:**

Use this formula: Length of the sector \(=(\frac{\theta}{180})πr\)

Length of the sector \(=(\frac{\theta}{180})πr=\) \((\frac{30}{180})π(20)=(\frac{1}{6})π(20)=(\frac{20}{6})×3.14 \cong 10.5\) \(cm\)

### Arc Length and Sector Area – Example 2:

Find the area of the sector. \((π=3.14)\), \(r=6 ft\) , \(\theta=70^\circ\)

**Solution:**

Use this formula: area of a sector \(=πr^2 (\frac{θ}{360}\))

Area of the sector \(=πr^2 (\frac{θ}{360})=(3.14)(6^2 )(\frac{70}{360})=(3.14)(36)(\frac {7}{36})= 21.98\) \(ft^2\)

### Arc Length and Sector Area – Example 3:

Find the length of the arc. \((π=3.14)\), \(r=3 ft\) , \(\theta=\frac{π}{3}\)

**Solution:**

Use this formula: Length of the sector \(=(\frac{\theta}{180})πr\)

\(\theta=\frac{π}{3}→\frac{π}{3}×\frac{180}{π}=60^\circ\)

Length of the sector\(=(\frac{θ}{180})πr\) \(=(\frac{60}{180})π(3)=(\frac{1}{3})π(3)=1×3.14=3.14\) \(ft\)

### Arc Length and Sector Area – Example 4:

Find the length of the arc. Round your answer to the nearest tenth. \((π=3.14)\) , \(r=8 cm\) ,\(\theta=30^\circ\)

**Solution:**

Use this formula: length of a sector\(=(\frac{θ}{180})πr\)

Length of a sector \(=(\frac{θ}{180})πr\) \(=(\frac{30}{180})π(8)=(\frac{1}{6})π(8)=1.3×3.14 \cong 4.2\) \(cm\)

## Exercises for Arc Length and Sector Area

### Arc Length and Sector Area.

Find the length of each arc. \((π=3.14)\)

1.\(\color{blue}{r=10 ft,θ=25^°}\)

2.\(\color{blue}{r=12 cm,θ=30^°}\)

3.\(\color{blue}{r=9 cm,θ=27^°}\)

4.\(\color{blue}{r=11 ft,θ=44^°}\)

Find area of each sector. \((π=3.14)\)

5.\(\color{blue}{r=5 ft,θ=40^°}\)

6.\(\color{blue}{r=6 cm,θ=60^°}\)

Length of the arc:

1.\(\color{blue}{4.36 ft}\)

2.\(\color{blue}{6.28 cm}\)

3.\(\color{blue}{4.24 cm}\)

4.\(\color{blue}{8.44 ft}\)

Area of a sector:

5.\(\color{blue}{8.72 ft^2}\)

6.\(\color{blue}{18.84 cm^2}\)

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