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Zorluk: ZorCircle Geometry: Arc Length and Sector Area

A circular stained-glass window with a radius of 12 inches12\text{ inches} contains a sector, OABOAB, with a central angle measuring 120∘120^\circ. A straight piece of lead wire is placed along chord AB‾\overline{AB}, dividing sector OABOAB into a triangular region △OAB\triangle OAB and a circular segment bounded by chord AB‾\overline{AB} and arc AB^\widehat{AB}. What is the area, in square inches, of the circular segment?

  1. 48π−36348\pi - 36\sqrt{3}Cevap
  2. B
    144π−363144\pi - 36\sqrt{3}
  3. C
    24π−36324\pi - 36\sqrt{3}
  4. D
    48π−72348\pi - 72\sqrt{3}
  5. E
    48π−3648\pi - 36

Cevap

48π−36348\pi - 36\sqrt{3} square inches
The area of a circular segment is found by subtracting the area of the central triangle from the area of the sector. The sector area with a 120∘120^\circ central angle and a radius of 1212 is 120∘360∘⋅π(122)=48π\frac{120^\circ}{360^\circ} \cdot \pi (12^2) = 48\pi. The area of the isosceles triangle formed by the two radii and the chord is 12(12)(12)sin⁡(120∘)=363\frac{1}{2} (12)(12) \sin(120^\circ) = 36\sqrt{3}. Thus, the segment area is 48π−36348\pi - 36\sqrt{3} square inches.

Adım Adım Çözüm

1
Calculate the area of sector OABOAB
Sector Area = 120∘360∘⋅π⋅(12)2=13⋅144π=48π sq in\frac{120^\circ}{360^\circ} \cdot \pi \cdot (12)^2 = \frac{1}{3} \cdot 144\pi = 48\pi\text{ sq in}
The area of a sector with central angle θ\theta in degrees is given by θ360∘πr2\frac{\theta}{360^\circ} \pi r^2.
2
Calculate the area of triangle △OAB\triangle OAB
Triangle Area = 12⋅12⋅12⋅sin⁡(120∘)=72⋅32=363 sq in\frac{1}{2} \cdot 12 \cdot 12 \cdot \sin(120^\circ) = 72 \cdot \frac{\sqrt{3}}{2} = 36\sqrt{3}\text{ sq in}
The area of a triangle with two sides aa and bb and included angle θ\theta is 12absin⁡(θ)\frac{1}{2} a b \sin(\theta).
3
Subtract the triangle area from the sector area to find the circular segment area
Segment Area = 48π−363 sq in48\pi - 36\sqrt{3}\text{ sq in}
A circular segment area is equal to the area of the containing sector minus the area of the central triangle.

Anahtar Kavram

Area of a Circular Segment
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Konular

Circle Geometry: Arc Length and Sector Area

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