Question

"in a square , all the mid points are joined. the inner square is shaded. if the area of the square is a, what is the shaded area?"

Answer 1

FULL ANSWER WITH STEPS AND EXPLANATION

The area of the square is $a$ square units. 
So, the side of the outer square should be $\sqrt{a}$ units. 
Each side of the outer square is divided into two equal segments by their corresponding midpoints. And, each side of the outer square measures $\sqrt{a}$ units. So, the measure of the half parts is $\frac{\sqrt{a}}{2}$ units. 
Now, if you make the figure containing the outer and inner squares, then you will also notice four congruent triangles formed. Two sides of one triangle would be equal, with both of them being $\frac{\sqrt{a}}{2}$ units in measure. 
Now, 
Area of the inner square (or shaded portion)
= Area of the outer square - Area of the four congruent triangles
$= a - [4 * (\frac{1}{2}*\frac{\sqrt{a}}{2}*\frac{\sqrt{a}}{2}]$
$= a - [4 * (\frac{1*\sqrt{a}*\sqrt{a}}{2*2*2}]$
$= a - [4 * \frac{a}{8}]$
$= a - \frac{a}{2}$
$= \frac{a}{2}$

∴ The area of the shaded portion should be $\frac{a}{2}$ square units. 

I tried my best to answer this question. 

Hope this may help you. 

If you have any doubt, then you can ask me in the comments.