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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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. 3.4.2: Deriving the Formula for the Area of a Sector

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

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

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

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. 3.4.2: Deriving the Formula for the Area of a Sector

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

✔ 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 http://www.walch.com/ei/00207 3.4.2: Deriving the Formula for the Area of a Sector

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

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

✔ 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 http://www.walch.com/ei/00208 3.4.2: Deriving the Formula for the Area of a Sector