Cosθ=OAAD​


cos30∘=23​​


AD=23​​×32


AD=163​cm


AB=323​cm


shaded region =  Area of circle - Area of triangle


                         =π(32)2−3×21​×323​×16


                         =(722578​−7683​)cm2

The area of the design can be found by subtracting the area of the equilateral triangle from the area of the circular table cover.

1. Find the area of the equilateral triangle:
- The equilateral triangle has all sides equal, and in this case, each side is equal to the radius of the circular table cover, which is 32 cm.
- The formula to find the area of an equilateral triangle is A = (√3/4) * s^2, where A is the area and s is the length of each side.
- Plugging in the values, we get A = (√3/4) * (32 cm)^2.
- Solving this equation, we find the area of the equilateral triangle to be approximately 277.95 cm^2.

2. Find the area of the circular table cover:
- The formula to find the area of a circle is A = π * r^2, where A is the area and r is the radius.
- Plugging in the values, we get A = π * (32 cm)^2.
- Solving this equation, we find the area of the circular table cover to be approximately 3216.99 cm^2.

3. Find the area of the design:
- To find the area of the design, we subtract the area of the equilateral triangle from the area of the circular table cover.
- A_design = A_cover - A_triangle.
- Plugging in the values, we get A_design = 3216.99 cm^2 - 277.95 cm^2.
- Solving this equation, we find the area of the design to be approximately 2939.04 cm^2.

Therefore, the area of the design formed by leaving an equilateral triangle in the middle of a circular table cover with a radius of 32 cm is approximately 2939.04 cm^2.