GATE Civil 2023 Set 1 | GA | Question: 3

Question

In the given figure, $\text{PQRSTV}$ is a regular hexagon with each side of length $5\mathrm{~cm}.$ A circle is drawn with its centre at $ \mathrm{V}$ such that it passes through $\mathrm{P}.$ What is the area $ \text{( in cm}^{2})$ of the shaded region? (The diagram is representative) 

• A. $\frac{25 \pi}{3}$ 
• B. $\frac{20 \pi}{3}$ 
• C. $6 \pi$ 
• D. $7\pi$ 

Comments

important point:

to find the angle of any polygon = [ ( total no of sides - 2 ) * 180 ] / no of sides

here the hexagon has 6 sides

so 4 * 180 / 6  = 120 degree

Answer 1

Sum of interior angles of a polygon = (n−2) × 180°

Each angle of a regular polygon = (n−2) × 180° / n

Hence, each angle of a regular hexagon will be 720°/6 = 120°

Area of the sector(i.e. shaded region) = (120°/360°) × π × 5² cm² = 25π/3 cm²

 

 

Comments

This should be the best answer.

Answer 2

Radius of circle = side of the hexagon = 5 cm.

Area of circle = 25 pi cm^2

Shaded region is 1/3 of the entire circle. Hence, area of shaded region = 25/3  pi cm^2

Comments

How do we know that "Shaded region is 1/3 of the entire circle."?

---

@Anant Singh_1

\[
\text{Given: side of regular hexagon} = 5 \text{ cm.}
\]

A regular hexagon can be divided into six equilateral triangles.  
Each equilateral triangle has internal angles of \(60^\circ\).

At vertex \(V\), two equilateral triangles are adjacent,  
hence the angle formed at \(V\) is
\[
60^\circ + 60^\circ = 120^\circ.
\]

The circle is drawn with center \(V\) and radius equal to the side of the hexagon, i.e. \(r = 5 \text{ cm.}\)

Now, the shaded region is a sector of the circle having a central angle of \(120^\circ\).

\[
\text{Area of shaded region} = \frac{120}{360} \times \pi r^2
\]

\[
= \frac{1}{3} \times \pi (5)^2
\]

\[
\boxed{\text{Area} = \frac{25\pi}{3} \text{ cm}^2.}
\]
 

---

Angle at centre is 360°, sector is covering 120°

120°/360° which is nothing bt 1/3

We are supposed to calculate area of shaded region, shaded region is 1/3 of whole circle

Answer 3

Area of sector = (pi * R * R * theta)/360 = (pi * 5 * 5 * 120) / 360 = (25 * pi) / 3

Option (a)

Answer 4

<PVT=120 DEGREE

FOR 360 DEGREE---→ AREA = PIE*5*5

FOR 120 DEGREEE------->X

CONVERTING IN RADIAN

X= PIE* 5 * 5 * (2*PIE/3 ) / 2*PIE

X=25 PIE/3