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Difficulty: HardCircle Geometry: Arc Length and Sector Area

In a circle with center OO, sector AOBAOB has an area of 18π18\pi square centimeters and an arc length along AB^\widehat{AB} of 3π3\pi centimeters. A straight line segment ABAB is drawn to complete triangle AOBAOB. What is the area, in square centimeters, of the circular segment bounded by line segment ABAB and arc AB^\widehat{AB}?

  1. 18π−36218\pi - 36\sqrt{2}Answer
  2. B
    18π−72218\pi - 72\sqrt{2}
  3. C
    18π−3618\pi - 36
  4. D
    36π−18236\pi - 18\sqrt{2}
  5. E
    18π−7218\pi - 72

Answer

The area of the circular segment is 18π−36218\pi - 36\sqrt{2} square centimeters.
The expression 18π−36218\pi - 36\sqrt{2} correctly represents the area of the circular segment. By dividing sector area (18π18\pi) by arc length (3π3\pi), we obtain 12r=6\frac{1}{2}r = 6, giving a radius r=12 cmr = 12\text{ cm}. Substituting r=12r = 12 into rθ=3πr\theta = 3\pi gives central angle θ=π4\theta = \frac{\pi}{4} radians (45∘45^\circ). The area of triangle AOBAOB is 12(12)2sin⁡(45∘)=362\frac{1}{2}(12)^2\sin(45^\circ) = 36\sqrt{2}. Subtracting the triangle area from the sector area yields 18π−36218\pi - 36\sqrt{2}.

Step-by-Step Solution

1
Relate sector area and arc length formulas to find radius rr
Sector area A=12r2θ=18πA = \frac{1}{2}r^2\theta = 18\pi and arc length s=rθ=3πs = r\theta = 3\pi. Dividing sector area by arc length gives 12r2θrθ=18π3π  ⟹  12r=6  ⟹  r=12 cm\frac{\frac{1}{2}r^2\theta}{r\theta} = \frac{18\pi}{3\pi} \implies \frac{1}{2}r = 6 \implies r = 12\text{ cm}.
Dividing the sector area equation by the arc length equation isolates the radius rr.
2
Find central angle θ\theta
s=rθ  ⟹  3π=12θ  ⟹  θ=3π12=π4 radianss = r\theta \implies 3\pi = 12\theta \implies \theta = \frac{3\pi}{12} = \frac{\pi}{4}\text{ radians} (45∘45^\circ).
Knowing the radius rr allows calculating θ\theta directly from the arc length formula.
3
Calculate the area of triangle AOBAOB
\text{Area}(AOB) = \frac{1}{2}r^2\sin\theta = \frac{1}{2}(12)^2\sin\left(\frac{\pi}{4}\right) = 72 \cdot \frac{\sqrt{2}}{2} = 36\sqrt{2}\text{ cm}^2.
The area of a triangle with two sides of length rr and included angle θ\theta is 12r2sin⁡θ\frac{1}{2}r^2\sin\theta.
4
Subtract the triangle area from the sector area to find the segment area
\text{Segment Area} = \text{Area}(\text{sector } AOB) - \text{Area}(\triangle AOB) = 18\pi - 36\sqrt{2}\text{ cm}^2.
The region bounded by the chord and the arc is the sector minus the central triangle.

Key Concept

The area of a circular segment is found by subtracting the area of the central triangle (12r2sin⁡θ\frac{1}{2}r^2\sin\theta) from the area of the circular sector (12r2θ\frac{1}{2}r^2\theta).
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Topics

Circle Geometry: Arc Length and Sector Area

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