Find the area of the shaded region in terms of pi.

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SUMMARY

The area of the shaded region in the given problem is calculated using the formula for the area of a sector, A = (θ/360)(πr²). For a circle with a radius of 12 meters, the area of the sector corresponding to a 120-degree angle is confirmed as 12π m², while the area for a 270-degree angle is calculated to be 108π m². The calculations are verified using the formula A = (1/2)r²θ, resulting in A = 108π m² for the 270-degree sector.

PREREQUISITES
  • Understanding of basic geometry concepts, specifically circles and sectors.
  • Familiarity with the formula for the area of a circle: A = πr².
  • Knowledge of how to convert degrees to radians for angular measurements.
  • Ability to perform calculations involving fractions and multiplication of constants.
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  • Study the derivation and applications of the area of a sector formula.
  • Learn how to convert between degrees and radians effectively.
  • Explore advanced geometry topics, such as the properties of circles and their sectors.
  • Practice solving problems involving areas of sectors with varying angles and radii.
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Etrujillo
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So far i have.

12) area of full circle is πr²
area of sector is (120/360)(πr²) or 12π

13) same
area is (270/360)(πr²)

Am i correct?

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#12 is correct ... finish #13
 
Im assuming 12m is the radius so \frac{270}{360}pi×12squared=108 pi
Am i correct?
 
$$A=\frac{1}{2}r^2\theta=\frac{1}{2}(12\text{ m})^2\frac{3\pi}{2}=108\pi\text{ m}^2\quad\checkmark$$
 

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