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Zorluk: KolayCircle Geometry: Arc Length and Sector Area

A circular garden plot has a radius of 10 meters10\text{ meters}. A sector of the garden with a central angle of 72∘72^\circ is planted with roses. What is the area, in square meters, of the sector planted with roses?

  1. A
    4π4\pi
  2. B
    40π40\pi
  3. 20π20\piCevap
  4. D
    72π72\pi
  5. E
    100π100\pi

Cevap

The correct answer is 20π20\pi, representing the area of the sector in square meters.
The area of a sector of a circle is calculated using the formula Area=θ360∘πr2\text{Area} = \frac{\theta}{360^\circ} \pi r^2. Substituting θ=72∘\theta = 72^\circ and r=10 metersr = 10\text{ meters} gives 72∘360∘π(10)2=15⋅100π=20π\frac{72^\circ}{360^\circ} \pi (10)^2 = \frac{1}{5} \cdot 100\pi = 20\pi.

Adım Adım Çözüm

1
Identify the formula for the area of a sector.
Area=θ360∘πr2\text{Area} = \frac{\theta}{360^\circ} \pi r^2
The area of a sector is a proportional fraction of the total area of the circle.
2
Substitute the given values into the formula.
Area=72∘360∘π(10)2\text{Area} = \frac{72^\circ}{360^\circ} \pi (10)^2
The central angle is 72∘72^\circ and the radius is 10 meters10\text{ meters}.
3
Simplify the fraction and the squared term.
Area=15π(100)\text{Area} = \frac{1}{5} \pi (100)
72/36072/360 simplifies to 1/51/5 and 102=10010^2 = 100.
4
Perform the final multiplication.
20π20\pi
One-fifth of 100100 is 2020.

Anahtar Kavram

Calculating the area of a circle sector given the radius and the central angle in degrees.
Tahmini Süre:1m 0s
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Konular

Circle Geometry: Arc Length and Sector Area

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