The area of an equilateral triangle ABC is 17320.5 cm2. With each vertex of the triangle as centre, a circle is drawn with radius equal to half the length of the side of the triangle (see Figure). Find the area of the shaded region. (Use π = 3.14 and √3 = 1.73205)
ABC is an equilateral triangle.
∴ ∠ A = ∠ B = ∠ C = 60°
There are three sectors each making 60°.
Area of ΔABC = 17320.5 cm2
⇒ √3/4 × (side)2 = 17320.5
⇒ (side)2 = 17320.5 × 4/1.73205
⇒ (side)2 = 4 × 104
⇒ side = 200 cm
Radius of the circles = 200/2 cm = 100 cm
Area of the sector = (60°/360°) × π r2 cm2
= 1/6 × 3.14 × (100)2 cm2
= 15700/3cm2
Area of 3 sectors = 3 × 15700/3 = 15700 cm2
Thus, area of the shaded region = Area of equilateral triangle ABC – Area of 3 sectors
= 17320.5 - 15700 cm2
= 1620.5 cm2