# How to Find Arc Length and Sector Area? (+FREE Worksheet!)

To find the Arc Length and Sector Area of a circle, you need the formulas and in this article, we review and explain these formulas and their application.

## 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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