Question 10
The area of an equilateral triangle ABC is 17320.5cm². 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 Fig. 12.28). Find the area of the shaded region. (Use π=3.14and√3=1.73205)

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Solution

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=200cm
Radius of the circles =200/2cm=100cm
Area of the sector =(60°/360°)×πr²cm²
=1/6×3.14×(100)²cm²
=15700/3cm²
Area of 3 sectors =3×15700/3=15700cm²
Area of the shaded region = Area of equilateral triangle ABC - Area of 3 sectors
=(17320.5−15700)cm²=1620.5cm²

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