Iam having trouble getting the correct ans of 7.16m
I understand your method, but the quoted $\displaystyle A=\pi~r^2$ formula only applies to full circles. It cannot be used to find an area of a segment, sector, etc. Or am I misinterpreting your logic? It seems to me that you are assuming that you can use the '0.5R' as a radius and the given area of the segment of 7.87 to find the radius of the whole circle.
1. You probably found the correct solution already. If not, here are the steps:
a) Calculate the value of the central angle of the sector:
$\displaystyle \cos\left(\frac12 \alpha\right=\dfrac{\frac12 R}{R}=\frac12~\implies~\frac12 \alpha = 60^\circ$. So the complete central angle is 120°.
b) The shaded area is a segment.
$\displaystyle area_{segment}=area_{sector} - area_{triangle}$
$\displaystyle 7.87 = \frac13 \cdot \pi R^2 - \frac12 \cdot R \cdot \underbrace{R \cdot \sin(120^\circ)}_{\text{height of triangle}}$
$\displaystyle 7.87 = \left(\frac13 \pi - \frac14 \sqrt{3} \right)R^2$
Solve for R.
c) the diameter is 2R.