# The area of an equilateral triangle ABC is 17320.5 cm^{2}. 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 = 3.14 and √3 = 1.73205)

**Solution:**

We use the formula for the area of the circle and the area of the triangle to solve the problem.

Given that, Area of equilateral Δ = 17320.5 cm^{2}

√3/4 (side)^{2} = 17320.5 cm^{2}

(side)^{2} = (17320.5 × 4)/√3 cm^{2}

= (17320.5 × 4)/1.73205 cm^{2}

side = √10000 × 4 cm²

= 100 × 2 cm

= 200 cm

Radius (r) = 1/2 × (length of side of triangle)

= 1/2 × 200 cm

= 100 cm

All interior angles of an equilateral traingle are of measure 60° and all 3 sectors are made using these interior angles.

∴ Angles subtended at the center by each sector (θ) = 60°

Area of each sector = θ/360° × πr^{2}

Area of 3 sectors = 3 × 60°/360° × πr^{2}

= 3 × 1/6 × 3.14 × (100 cm)^{2}

= 15700 cm^{2}

Area of shaded region = Area of ΔABC - Area of 3 sectors

= 17320.5 cm^{2} - 15700 cm^{2}

= 1620.5 cm^{2}

**Video Solution:**

## 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 Fig. 12.28). Find the area of the shaded region

### NCERT Solutions Class 10 Maths - Chapter 12 Exercise 12.3 Question 10:

The area of an equilateral triangle ABC is 17320.5 cm^{2}. 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

The area of the shaded region if a circle is drawn with each vertex of the equilateral triangle as center is 1620.5 cm^{2}.