Question

A hexagonal field of side 30 m is to be used for playing cricket. As circular fields are preferred for cricket, a circle of radius 30 m is made circumscribing the hexagon so as to make the field circular. Find the increase in the area of the cricket field.

• A. 387.74 $m^2$
• B. 487.74 $m^2$
• C. 587.7 $m^2$
• D. 667.6 $m^2$

Answer 1

The correct option is B

487.74 $m^2$

A hexagon contains six equilateral triangles, the length of the side of the triangle is equal to the length of the side of the hexagon. In the above question, the length of the side of the hexagon is 30 m, which means that the length of the sides of the equilateral triangle would be 30 m.

Using Heron’s formula to calculate the area of an equilateral triangle of side 30 m.
Semi-perimeter = $\frac{(30+30+30)}{2}$ = 45 m.
Heron's formula says,
A =$\sqrt{s(s-a)(s-b)(s-c)}$
= $\sqrt{45(45-30)(45-30)(45-30)}$
= $\sqrt{45\times 15\times 15\times 15}$
=$15\times 15\times \sqrt{3}$
= 389.71 $m^2$
Area of the hexagonal field = 6$\times$389.71 = 2338.26 $m^2$.
Now, a circle of radius 30 m is circumscribing the hexagonal field,
Area of the circle = 3.14$\times$${30}^2$ = 2826 $m^2$.

So, the increase in area = 2826 $m^2$ - 2338.26 $m^2$ = 487.74 $m^2$.

[Alternatively, area(A) of the equilateral triangle can be directly calculated using the formula, $A=\frac{\sqrt{3}}{4}{a}^2$, where 'a' is its side.]