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slide1

Introduction

A sector is the portion of a circle bounded by two radii and their intercepted arc. Previously, we thought of arc length as a fraction of the circumference of the circle. In a similar way, we can think of a sector as a fraction of the area of the circle. In the same way that we found arc length, we can set up proportions to find the area of a sector.

3.4.2: Deriving the Formula for the Area of a Sector

key concepts a sector is the portion of a circle bounded by two radii and their intercepted arc
Key Concepts

A sector is the portion

of a circle bounded

by two radii and their

intercepted arc.

3.4.2: Deriving the Formula for the Area of a Sector

slide3

Key Concepts, continued

To find the area of a sector, , when the central angle θis given in radians, we can set up a proportion using the area of a circle,

We can solve this proportion for the area of the sector and simplify to get a formula for the area of a sector in terms of the radius of the circle and the radian measure of the central angle θ.

3.4.2: Deriving the Formula for the Area of a Sector

slide4

Key Concepts, continued

To find the area of a sector when the central angle is given in degrees, we can set up a proportion using the area of a circle.

3.4.2: Deriving the Formula for the Area of a Sector

slide5
Common Errors/Misconceptions

inconsistently setting up ratios

incorrectly simplifying ratios involving

3.4.2: Deriving the Formula for the Area of a Sector

slide6
Guided Practice

Example 1

A circle has a radius of 24

units. Find the area of a

sector with a central

angle of 30°.

3.4.2: Deriving the Formula for the Area of a Sector

guided practice example 1 continued find the area of the circle
Guided Practice: Example 1, continued

Find the area of the circle.

3.4.2: Deriving the Formula for the Area of a Sector

guided practice example 1 continued set up a proportion
Guided Practice: Example 1, continued

Set up a proportion.

3.4.2: Deriving the Formula for the Area of a Sector

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Guided Practice: Example 1, continued

Multiply both sides by the area of the circle to find the area of the sector.

The area of the sector is approximately 150.80 units2.

3.4.2: Deriving the Formula for the Area of a Sector

guided practice example 1 continued
Guided Practice: Example 1, continued

3.4.2: Deriving the Formula for the Area of a Sector

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Guided Practice

Example 3

A circle has a radius of 6

units. Find the area of a

sector with an arc length

of 9 units.

3.4.2: Deriving the Formula for the Area of a Sector

slide12
Guided Practice: Example 3, continued

Use the radian measure formula to find the measure of the central angle.

3.4.2: Deriving the Formula for the Area of a Sector

slide13

Guided Practice: Example 3, continued

Substitute radius and radian measure into the formula for the area of a sector and simplify.

The area of the sector is 27 units2.

3.4.2: Deriving the Formula for the Area of a Sector

guided practice example 3 continued
Guided Practice: Example 3, continued

3.4.2: Deriving the Formula for the Area of a Sector