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Zorluk: OrtaCircle Geometry: Arc Length and Sector Area

In a circle with center OO, a sector has a central angle measuring 2π5\frac{2\pi}{5} radians and an arc length of 6π6\pi centimeters. What is the area, in square centimeters, of the sector?

  1. A
    15π15\pi
  2. B
    9π5\frac{9\pi}{5}
  3. 45π45\piCevap
  4. D
    90π90\pi
  5. E
    225π225\pi

Cevap

The area of the sector is 45π45\pi square centimeters.
To find the area of the sector, first determine the radius rr of the circle using the arc length formula for radians, s=rθs = r\theta. Substituting s=6πs = 6\pi and θ=2π5\theta = \frac{2\pi}{5} yields 6π=r(2π5)6\pi = r \left(\frac{2\pi}{5}\right), which simplifies to r=15r = 15 centimeters. Then, apply the sector area formula A=12r2θA = \frac{1}{2} r^2 \theta. Substituting r=15r = 15 and θ=2π5\theta = \frac{2\pi}{5} gives A=12(152)(2π5)=45πA = \frac{1}{2} (15^2) \left(\frac{2\pi}{5}\right) = 45\pi square centimeters. Alternatively, using A=12rsA = \frac{1}{2} r s directly gives A=12(15)(6π)=45πA = \frac{1}{2} (15)(6\pi) = 45\pi square centimeters.

Adım Adım Çözüm

1
Find the radius of the circle using the arc length formula in radians.
r=15r = 15 centimeters
The arc length formula for an angle in radians is s=rθs = r\theta. Substituting s=6πs = 6\pi and θ=2π5\theta = \frac{2\pi}{5} gives 6π=r(2π5)6\pi = r \left(\frac{2\pi}{5}\right), which yields r=6π⋅52π=15r = 6\pi \cdot \frac{5}{2\pi} = 15 cm.
2
Calculate the area of the sector using the sector area formula.
A=45πA = 45\pi square centimeters
The area of a sector with central angle θ\theta in radians is A=12r2θA = \frac{1}{2} r^2 \theta. Substituting r=15r = 15 and θ=2π5\theta = \frac{2\pi}{5} gives A=12(152)(2π5)=12(225)(2π5)=45πA = \frac{1}{2} (15^2) \left(\frac{2\pi}{5}\right) = \frac{1}{2} (225) \left(\frac{2\pi}{5}\right) = 45\pi square centimeters.

Anahtar Kavram

Arc Length and Sector Area in Radians
Tahmini Süre:1m 0s
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Konular

Circle Geometry: Arc Length and Sector Area

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