What is the area of the segment formed by a chord in a circle of radius 12 cm, if the angle subtended at the centre is 150°?
60π − 36
Area of a (minor) segment = Area of sector − Area of triangle formed by the two radii and the chord.
Step 1 – Sector area with \(\theta = 150^\circ, r = 12\):
\(A_{\text{sector}} = \dfrac{\theta}{360^\circ} \times \pi r^2 = \dfrac{150}{360} \times \pi \times 144 = \dfrac{5}{12} \times 144\pi = 60\pi\)
Step 2 – Triangle area with two sides \(r = 12\) and included angle \(150^\circ\):
\(A_{\triangle} = \dfrac{1}{2} r^2 \sin\theta = \dfrac{1}{2} \times 144 \times \sin 150^\circ = \dfrac{1}{2} \times 144 \times \dfrac{1}{2} = 36\)
Step 3 – Segment area:
\(A_{\text{seg}} = 60\pi - 36\)
Hence the correct answer is option 1) 60π − 36.
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