The circumference of a circle is `100` cm. The side of a square inscribed in the circle is :

Answer

To find the side of a square inscribed in a circle with a circumference of 100 cm, we can follow these steps:

### Step 1: Find the radius of the circle
The formula for the circumference \( C \) of a circle is given by:
\[ C = 2\pi r \]
where \( r \) is the radius of the circle. We can rearrange this formula to solve for \( r \):
\[ r = \frac{C}{2\pi} \]
Substituting the given circumference:
\[ r = \frac{100}{2\pi} = \frac{50}{\pi} \text{ cm} \]

### Step 2: Relate the radius to the side of the inscribed square
The diagonal \( d \) of the inscribed square is equal to the diameter of the circle. The diameter \( D \) can be calculated as:
\[ D = 2r \]
Substituting the value of \( r \):
\[ D = 2 \times \frac{50}{\pi} = \frac{100}{\pi} \text{ cm} \]

### Step 3: Use the relationship between the side of the square and its diagonal
For a square with side length \( s \), the relationship between the side length and the diagonal is given by:
\[ d = s\sqrt{2} \]
Setting the diagonal equal to the diameter of the circle:
\[ s\sqrt{2} = \frac{100}{\pi} \]

### Step 4: Solve for the side length \( s \)
To find \( s \), we can rearrange the equation:
\[ s = \frac{100}{\pi\sqrt{2}} \]

### Final Step: Simplify the expression
This gives us the side length of the inscribed square:
\[ s = \frac{100}{\pi\sqrt{2}} \text{ cm} \]

### Summary
The side of the square inscribed in the circle is:
\[ s = \frac{100}{\pi\sqrt{2}} \text{ cm} \]