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Zorluk: OrtaCircle Geometry: Arc Length and Sector Area

The tip of a pendulum swings through a central angle measuring π3\frac{\pi}{3} radians, tracing an arc length of 4π4\pi centimeters. What is the area, in square centimeters, of the sector swept out by the pendulum?

  1. A
    12π12\pi
  2. 24π24\piCevap
  3. C
    36π36\pi
  4. D
    48π48\pi
  5. E
    144π144\pi

Cevap

24π24\pi square centimeters
To find the area of the sector, first determine the radius of the pendulum's circular path using the arc length formula s=rθs = r\theta. Substituting s=4πs = 4\pi and θ=π3\theta = \frac{\pi}{3} gives r=12r = 12 cm. Then, substitute the radius and central angle into the sector area formula A=12r2θ=12(12)2(π3)=24πA = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2\left(\frac{\pi}{3}\right) = 24\pi square centimeters.

Adım Adım Çözüm

1
Calculate the radius of the circle using the arc length formula in radians.
Using s=rθs = r\theta with s=4πs = 4\pi and θ=π3\theta = \frac{\pi}{3} gives 4π=r(π3)  ⟹  r=124\pi = r\left(\frac{\pi}{3}\right) \implies r = 12 cm.
The radius is required to calculate the sector area.
2
Calculate the area of the sector using the radian sector area formula.
A=12r2θ=12(12)2(π3)=12(144)(π3)=24πA = \frac{1}{2}r^2\theta = \frac{1}{2}(12)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(144)\left(\frac{\pi}{3}\right) = 24\pi square centimeters.
Multiplying half the squared radius by the radian measure of the central angle yields the sector area.

Anahtar Kavram

Arc Length and Sector Area in Radians
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Konular

Circle Geometry: Arc Length and Sector Area

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