A circle has a diameter of 20 inches. Find the area of the sector with the following central angles: 55°, 20°, 62°.
Answer (2)
Calculate the central angle for Sector 1 using the sector area formula: θ 1 = π r 2 A re a 1 × 360 = 15. 5 ∘ .
Determine the area for Sector 2 using the sector area formula: A re a 2 = 360 θ 2 π r 2 = 2 25 π .
Compute the central angle for Sector 3 using the sector area formula: θ 3 = π r 2 A re a 3 × 360 = 11.2 5 ∘ .
State the missing values: A re a 2 = 2 25 π , θ 1 = 15. 5 ∘ , θ 3 = 11.2 5 ∘ .
Explanation
Problem Analysis We are given a circle with a diameter of 20 inches, which means its radius is 10 inches. We have three sectors with either the area or the central angle given. Our goal is to find the missing values.
Finding Central Angle for Sector 1 For Sector 1, we are given the area 36 155 π square inches. We need to find the central angle. The formula for the area of a sector is: A re a = 360 θ π r 2 where θ is the central angle in degrees and r is the radius. Plugging in the given values: 36 155 π = 360 θ π ( 10 ) 2 Solving for θ :
θ = π × 100 36 155 π × 360 = 36 × 100 155 × 360 = 10 155 = 15. 5 ∘ So, the central angle for Sector 1 is 15. 5 ∘ .
Finding Area for Sector 2 For Sector 2, we are given the central angle 4 5 ∘ . We need to find the area. Using the same formula: A re a = 360 θ π r 2 Plugging in the given values: A re a = 360 45 π ( 10 ) 2 = 8 1 π ( 100 ) = 8 100 π = 2 25 π So, the area for Sector 2 is 2 25 π square inches.
Finding Central Angle for Sector 3 For Sector 3, we are given the area 8 25 π square inches. We need to find the central angle. Using the formula: A re a = 360 θ π r 2 Plugging in the given values: 8 25 π = 360 θ π ( 10 ) 2 Solving for θ :
θ = π × 100 8 25 π × 360 = 8 × 100 25 × 360 = 2 × 5 25 × 9 = 2 5 × 9 = 2 45 = 22. 5 ∘ So, the central angle for Sector 3 is 11.2 5 ∘ .
Final Answer Therefore, the missing central angle for Sector 1 is 15. 5 ∘ , the missing area for Sector 2 is 2 25 π square inches, and the missing central angle for Sector 3 is 11.2 5 ∘ .
Examples
Understanding sectors and their areas is crucial in many real-world applications. For instance, when designing a sprinkler system for a circular lawn, you need to calculate the area each sprinkler head will cover to ensure complete and efficient watering. Similarly, in baking, if you're cutting a circular pie into slices, knowing the central angle and radius helps determine the area of each slice, ensuring fair distribution. These calculations also come into play in fields like optics, where understanding the area covered by a lens's field of view is essential.
The areas of the sectors in a circle with a diameter of 20 inches for the angles 55°, 20°, and 62° are approximately 48.99, 17.45, and 54.63 square inches, respectively. These areas are calculated using the formula for the area of a sector. Understanding this allows us to analyze parts of circles effectively.
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