In the given figure, PQRSTV is a regular hexagon with each side of length 5 cm. A circle is drawn with its centre at V such that it passes through P. What is the area (in cm^2) of the shaded region? (The diagram is representative)
A. 25pi/3
B. 20pi/3
C. 6pi
D. 7pi
GATE 2023 · General Aptitude · Area · medium
Answer: Area of shaded region = 25pi/3 cm^2
Identify the radius of the circle: In hexagon PQRSTV the vertices V and P are consecutive (adjacent). Adjacent vertices are separated by one side. So VP = s = 5 cm. The circle centred at V passes through P, hence radius r = 5 cm.
Determine the sector angle at V: For n = 6: interior angle = (6-2)*180/6 = 4*180/6 = 120 degrees. The sector at V spans from radius VP to radius VT (the two sides meeting at V), so the sector angle theta = 120 degrees.
Compute the sector area: Area = (120/360) * pi * (5)^2 = (1/3) * pi * 25 = 25pi/3 cm^2.