Question

You are working for the company Trinity Mathematical Concepts are

asked for create the following logo for the company. If each vertices of the triangle are the centers of the circles, what is the area of the shaded region?
Your answer will be in terms of "x", where "x" is the radius of the circles.

Answer 1

• 
  `(2\pi -3√(3))x^2/4`

Explanation:

The sides of the triangle in the middle are equal to x so tis is equilateral triangle.

Its interior angle is 60°.

Each shaded part is the area of 60° segment of circle with the radius x.

The area of each segment is the difference between 60° sector and the area of triangle.

• 
  `A_(segment)=A_(sector)-A_(triangle)`

Area of sector

• 
  `A_(sector) = (60/360)*\pi r^2=(1/6)\pi x^2`

Area of triangle

• 
  `A_(triangle)=(√(3)/4)a^2= (√(3)/4)x^2`

The shaded area is

• 
  `A = 3((1/6)\pi x^2 - (√(3) /4)x^2)=((1/2)\pi -3√(3) /4)x^2=(2\pi -3√(3))x^2/4`