In the figure shown above, each inside square is formed by joining the mid points of the sides of the next larger square. The area of the smallest square (shaded) as shown, in cm² is

The correct option is 3.125.

Step 1: Understand the geometric relationship between the nested squares
Let the side length of a square be s. The area of this square is given by:

$$A=s^2$$
When we join the midpoints of the sides of this square to form a new, inscribed square, the vertices of the new square divide each side of the original square into two halves, each of length s/2.
By the Pythagorean theorem, the side length of the new inscribed square, s_new, is the hypotenuse of a right-angled triangle with legs of length s/2:

$$s_{\text{new}}=\sqrt{{\left(\frac{s}{2}\right)}^2+{\left(\frac{s}{2}\right)}^2}=\sqrt{\frac{s^2}{4}}=\sqrt{\frac{s^2}{2}}=\frac{s}{\sqrt{2}}$$
The area of this new inscribed square, A_new, is:

$$A_{\text{new}}={\left(s_{\text{new}}\right)}^2={\left(\frac{s}{\sqrt{2}}\right)}^2=\frac{s^2}{2}=\frac{A}{2}$$
Thus, the area of each successive square formed by joining the midpoints of the sides is exactly half the area of the preceding square.

Step 2: Calculate the area of each square step-by-step
From the labels in the provided image, the outermost square (1st square) has a side length of 10 cm.
Let A₁ be the area of the 1st square:

$$A_1=10\text{cm}\times 10\text{cm}=100{\text{cm}}^2$$
Now, we calculate the area of each subsequent nested square:

• 2nd Square:
  $A_2=\frac{A_1}{2}=\frac{100}{2}=50{\text{cm}}^2$


• 3rd Square:
  $A_3=\frac{A_2}{2}=\frac{50}{2}=25{\text{cm}}^2$


• 4th Square:
  $A_4=\frac{A_3}{2}=\frac{25}{2}=12.5{\text{cm}}^2$


• 5th Square:
  $A_5=\frac{A_4}{2}=\frac{12.5}{2}=6.25{\text{cm}}^2$


• 6th Square (Innermost, shaded square):
  $A_6=\frac{A_5}{2}=\frac{6.25}{2}=3.125{\text{cm}}^2$

Therefore, the area of the smallest shaded square is 3.125 cm².