GATE 2023 CE (CE1) – Question 3

General Aptitude · Quantitative Aptitude: Algebra, Geometry and Mensuration · 1 mark · Multiple choice

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 regular hexagon PQRSTV with the circle of radius VP = 5 cm centred at V. The shaded region is the part of the circle inside the hexagon, between the radii VP and VT.

$\frac{25\pi}{3}$
$\frac{20\pi}{3}$
$6\pi$
$7\pi$

Practise this question in The GATE Grind →

Show answer and explanation

Correct answer: (A) $\frac{25\pi}{3}$

Explanation

The radius is $VP = 5$ cm. The interior angle of a regular hexagon is $120°$, so the shaded part is the sector of the circle of angle $\angle PVT = 120°$ (the arc from P to T lies inside the hexagon). Its area is $\frac{120}{360}\pi \times 5^2 = \frac{25\pi}{3}$ cm².