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Difficulty: MediumCircle Geometry: Arc Length and Sector Area

A circular section of a community park is bounded by two straight footpaths meeting at the park's center at a 75∘75^\circ angle, and an outer curved walking trail. If the length of the outer curved walking trail is 5π5\pi meters, what is the area, in square meters, of this section of the park?

  1. A
    25π25\pi
  2. 30π30\piAnswer
  3. C
    60π60\pi
  4. D
    120π120\pi
  5. E
    144π144\pi

Answer

The area of this section of the park is 30π30\pi square meters.
The central angle of 75∘75^\circ represents 75360=524\frac{75}{360} = \frac{5}{24} of the circle. Setting the arc length 5π=524(2πr)5\pi = \frac{5}{24} (2\pi r) gives 5πr12=5π\frac{5\pi r}{12} = 5\pi, which simplifies to r=12r = 12 meters. The sector area is 524⋅π(12)2=30π\frac{5}{24} \cdot \pi (12)^2 = 30\pi square meters.

Step-by-Step Solution

1
Find the radius of the circular park section using the arc length formula.
r=12r = 12 meters
The arc length formula is s=θ360∘⋅2πrs = \frac{\theta}{360^\circ} \cdot 2\pi r. Substituting s=5πs = 5\pi and θ=75∘\theta = 75^\circ gives 5π=75360⋅2πr  ⟹  5π=524⋅2πr  ⟹  5π=5πr12  ⟹  r=125\pi = \frac{75}{360} \cdot 2\pi r \implies 5\pi = \frac{5}{24} \cdot 2\pi r \implies 5\pi = \frac{5\pi r}{12} \implies r = 12 meters.
2
Calculate the sector area using the radius and central angle.
Area = 30π30\pi square meters
The sector area formula is A=θ360∘⋅πr2A = \frac{\theta}{360^\circ} \cdot \pi r^2. Substituting θ=75∘\theta = 75^\circ and r=12r = 12 gives A=75360⋅π(12)2=524⋅144π=30πA = \frac{75}{360} \cdot \pi (12)^2 = \frac{5}{24} \cdot 144\pi = 30\pi square meters.

Key Concept

Determining Sector Area from Central Angle and Arc Length
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Topics

Circle Geometry: Arc Length and Sector Area

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