Question

ABC is an equilateral triangle of side 8 cm. A, B and C are the centres of circular arcs of radius 4 cm. Find the area of the shaded region correct upto 2 decimal places.(Take $\pi =3.142$ and $\sqrt{3}=1.732$)

• A. 4 $c{m}^2$
• B. 2.59 $c{m}^2$
• C. 1.39 $c{m}^2$
• D. 6 $c{m}^2$

Answer 1

The correct option is B

2.59 $c{m}^2$

The area of the shaded section is equal to the area of the triangle minus the areas of the three sectors. Since it is an equilateral triangle, each sector is one sixth of the area of the circle with radius 4, which means the area of all three sectors is one half the area of the circle.

The area of a triangle is $\frac{1}{2}\times base\times height$,

base = 8 cm and height = $\frac{\sqrt{3}\times 8}{2}$ cm (height of an equilateral triangle is $\frac{\sqrt{3}\times a}{2}$

Area = $\frac{1}{2}\times 8\times \frac{\sqrt{3}\times 8}{2}$

= $16\sqrt{3}$ = 27.71 $c{m}^2$

Area of 3 sectors = $\frac{1}{2}\times \pi \times 4^2$

= $3.14\times 8$ = 25.12 $c{m}^2$

Now area of shaded portion = 27.71 - 25.12 = 2.59 $c{m}^2$