How to calculate perimeter using the area and the perimeter of a smaller area 
I have having trouble understanding how to break this problem apart.
I have an $ L$ shape with a rectangle in it.
The smaller rectangle has a side of $5 m$ and a side of $7 m$, the $L$ shape has an area of $6.25m^2$
I can work out the total area for both but not how to use the area to calculate the perimeter of the $L$ shape when no sides length or width are know.

The geometric problem is not fully specified.  We assume that the vertical and horizontal arms of the L have the same thickness $d$. (We were not told that the thicknesses are the same.) We suppose also that the $5\times 7$ rectangle referred to is the rectangle embraced by the two arms of the L, but outside the L.  
Then if we draw the L shape, the little $d \times d$ square in the lower left corner has area $d^2$, and the rest of the L has area $d(5+7)$. So we obtain the equation
$$d^2+12d=6.25.$$
Rewrite as $d^2+12d-6.25=0$, and solve. The positive root is $1/2$.
Now the perimeter of the L is easy to find.  It is $2(7) +2(5)+4(1/2)$.
Remark: Is the above the right interpretation? That is certainly not clear. But the happy fact that the discriminant $(12)^2-(4)(-6.25)$ is a perfect square is evidence of sorts that the interpretation might be the intended one. 
