The perimeter of a certain sector of a circle of radius 6.5 cm in 31 cm.

Given:
Radius = 6.5 cm

Let O be the centre of the circle with radius 6.5 cm and OACBO be its sector with perimeter 31 cm.
Thus, we have:
OA + OB + arc AB = 31 cm

$\Rightarrow 6.5+6.5+\operatorname{arc} A B=31$

$\Rightarrow \operatorname{arc} A B=31-13$

$\Rightarrow \operatorname{arc} A B=18 \mathrm{~cm}$

Now,

Area of the sector $\mathrm{OACBO}=\frac{1}{2} \times$ Radius $\times$ Arc

$=\frac{1}{2} \times 6.5 \times 18$

$=58.5 \mathrm{~cm}^{2}$