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Zorluk: KolayCircle Geometry: Arc Length and Sector Area

A board game spinner has a pointer of length 6 inches6\text{ inches} that pivots at the center of the circular board. If the sector of the board representing 'Lose a Turn' has a central angle of π3\frac{\pi}{3} radians, what is the area, in square inches, of this sector?

  1. A
    2π2\pi
  2. B
    3π2\frac{3\pi}{2}
  3. 6π6\piCevap
  4. D
    12π12\pi
  5. E
    36π36\pi

Cevap

6π6\pi square inches
The area of a sector with a radius rr and a central angle θ\theta (measured in radians) is determined by the formula A=12r2θA = \frac{1}{2}r^2\theta. Given that the radius r=6r = 6 and the central angle θ=π3\theta = \frac{\pi}{3}, substituting these values into the formula gives A=12(6)2(π3)=12(36)(π3)=6πA = \frac{1}{2}(6)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(36)\left(\frac{\pi}{3}\right) = 6\pi square inches.

Adım Adım Çözüm

1
Identify the radius and the central angle of the sector from the problem description.
Radius r=6 inchesr = 6\text{ inches}, and central angle θ=π3 radians\theta = \frac{\pi}{3}\text{ radians}.
These values are the direct inputs required for the circle sector formulas.
2
Use the sector area formula in radians, A=12r2θA = \frac{1}{2}r^2\theta, to calculate the area.
A=12(6)2(π3)=12(36)(π3)=18(π3)=6πA = \frac{1}{2}(6)^2\left(\frac{\pi}{3}\right) = \frac{1}{2}(36)\left(\frac{\pi}{3}\right) = 18\left(\frac{\pi}{3}\right) = 6\pi.
This formula scales the total area of the circle by the fraction represented by the radian angle relative to 2π2\pi radians.

Anahtar Kavram

Calculating the area of a circle sector using radian measure
Tahmini Süre:1m 0s
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Konular

Circle Geometry: Arc Length and Sector Area

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