Step-by-step Solution:

To find the required area, we subtract the area of the three circular sectors from the triangle's total area.

1. Find the side of the triangle (a):
The area of an equilateral triangle is \(A = \frac{\sqrt{3}}{4}a^2\).
Given \(A = 49\sqrt{3} \text{ cm}^2\).
\[\begin{aligned} 49\sqrt{3} &= \frac{\sqrt{3}}{4}a^2 \\ 49 \times 4 &= a^2 \\ a &= \sqrt{196} = 14 \text{ cm} \end{aligned}\]
2. Find the radius of the circles (r):
The radius is half the side length.
\[r = \frac{a}{2} = \frac{14}{2} = 7 \text{ cm}\]
3. Find the area of the three sectors:
The three \(60^\circ\) sectors at the vertices combine to form a semicircle (\(3 \times 60^\circ = 180^\circ\)).
Area of three sectors = Area of a semicircle.
\[A_{\text{sectors}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times \frac{22}{7} \times (7)^2 = 77 \text{ cm}^2\]
4. Calculate the final area:
First, find the numerical area of the triangle using \(\sqrt{3} = 1.73\).
\[A_{\text{triangle}} = 49\sqrt{3} = 49 \times 1.73 = 84.77 \text{ cm}^2\]
Now, subtract the area of the sectors from the area of the triangle.
\[A_{\text{required}} = A_{\text{triangle}} - A_{\text{sectors}} = 84.77 - 77 = 7.77 \text{ cm}^2\]
Result:
The area of the triangle not included in the circles is \(7.77 \text{ cm}^2\).