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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.

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Presentation on theme: "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."— Presentation transcript:

1 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

2 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

3 Key Concepts, continued
To find the area of a sector, , when the central angle θ is 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

4 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

5 Common Errors/Misconceptions
inconsistently setting up ratios incorrectly simplifying ratios involving 3.4.2: Deriving the Formula for the Area of a Sector

6 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

7 Guided Practice: Example 1, continued Find the area of the circle.
3.4.2: Deriving the Formula for the Area of a Sector

8 Guided Practice: Example 1, continued Set up a proportion.
3.4.2: Deriving the Formula for the Area of a Sector

9 ✔ 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 units2. ✔ 3.4.2: Deriving the Formula for the Area of a Sector

10 Guided Practice: Example 1, continued
3.4.2: Deriving the Formula for the Area of a Sector

11 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

12 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

13 ✔ 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

14 Guided Practice: Example 3, continued
3.4.2: Deriving the Formula for the Area of a Sector

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