Question

Tilak draw a triangular design. The triangle  he drawn is equilateral with side 12 cm and then with the help of bangle he draw a circle inscribed in it touching its sides. Calculate the remaining area of triangle which is untouched by a circle.
1. 20.4 cm²
2. 26.4 cm2
3. 24.6 cm2
4. 22.6 cm2

Answer 1

Correct Answer - Option 3 : 24.6 cm2

Given∶

An equilateral triangle with side 12 cm.

Formula Used∶

Area of an equilateral triangle = (√3/4) × side²

Height of an equilateral triangle = (√ 3/2) × side 

Calculation ∶

When a circle is inscribed in an equilateral triangle, the radius of the circle will be 1/3 of the height of the triangle.

So, Height of an equilateral triangles = (√3/2) × side

h = √3/2 × 12 = 6√3 cm.

Now, Radius of the  circle = 1/3 × height

⇒ r = 1/3 × 6√3 = 2√3 cm

Area of an equilateral triangle = √3/4 side²

⇒ A = √3/4 × (12)² = 36√3 cm²

Now, Area of the circle = πr²

⇒ a = π × (2√3)²

⇒ a = 12 π cm²

Area of the remaining portion of the triangle = A - a = 36√3 - 12π  = 36 × 1.73 - 12 × 3.14 = 62.28 - 37.68 = 24.6 cm²

∴ The area of the remaining portion of the triangle is 24.6 cm².