The difference between the area of the circumscribed circle and the area of the inscribed circle of an equilateral triangle is 2156 sq. cm. What is the area of the equilateral triangle?
Explanation:
Let ‘a’ be the side of the triangle.
Circum-radius of an equilateral triangle = a/√3 ∴ Area of the circum-circle = π × (a/√3)2 = πa2/3
In-radius of an equilateral triangle = a/2√3 ∴ Area of the in-circle = π × (a/2√3)2 = πa2/12
According to the question,
πa2/3 - πa2/12 = 2156
⇒ πa2/4 = 2156
⇒ a2 = 2156 × 4 × 7/22 = 2744
∴ Area of the equilateral triangle = √3/4 × a2 = √3/4 × 2744 = 686√3.
Hence, option (a).
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