An equilateral triangle is inscribed in a circle of radius 1 unit. The area of the shaded region, in square unit, is:
π/3 - √3/4
π/3 - √3/2
π - 3
π - 3/4
Answer is Wrong!
Answer is Right!
This question was previously asked in
UPSC CAPF – 2018
An equilateral triangle inscribed in a circle of radius 1 unit has sides that subtend a central angle of 360°/3 = 120°. The area of a sector of the circle corresponding to this angle is (120°/360°) * π * r² = (1/3) * π * 1² = π/3. The area of the triangle formed by the center of the circle and one side of the equilateral triangle is (1/2) * r * r * sin(120°) = (1/2) * 1 * 1 * (√3/2) = √3/4. The region between the arc and the chord (one segment of the circle) has an area equal to the area of the sector minus the area of this triangle: π/3 – √3/4. Assuming the shaded region in the missing diagram refers to one such segment, this is the correct area. If the shaded area referred to the total area of the circle outside the triangle (three segments), the area would be 3 * (π/3 – √3/4) = π – 3√3/4, which is not among the options. Therefore, it is highly likely that the shaded region depicted was one segment.
For an equilateral triangle inscribed in a circle, each side subtends a central angle of 120°. The area of a circular segment formed by a chord and its corresponding arc is the area of the sector minus the area of the triangle formed by the radii and the chord.