The circumference of a circle is 25 cm. Find the side of the square inscribed in the circle.

Options

A. `25/(pisqrt(2)) cm`
B. `21/(pisqrt(3)) cm`
C. `23/(pisqrt(2)) cm`
D. `29/(pisqrt(3)) cm`

Answer

To find the side of the square inscribed in a circle with a circumference of 25 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:
\[ C = 2\pi r \]
Given that the circumference \( C = 25 \) cm, we can set up the equation:
\[ 25 = 2\pi r \]
To find the radius \( r \), we rearrange the equation:
\[ r = \frac{25}{2\pi} \]

### Step 2: Find the diameter of the circle
The diameter \( d \) of the circle is twice the radius:
\[ d = 2r \]
Substituting the value of \( r \):
\[ d = 2 \times \frac{25}{2\pi} = \frac{25}{\pi} \]

### Step 3: Relate the diameter of the circle to the diagonal of the square
The diagonal \( D \) of the inscribed square is equal to the diameter of the circle:
\[ D = d = \frac{25}{\pi} \]

### Step 4: Relate the diagonal of the square to its side length
For a square with side length \( a \), the relationship between the diagonal \( D \) and the side length \( a \) is given by:
\[ D = a\sqrt{2} \]
Setting the two expressions for the diagonal equal gives:
\[ a\sqrt{2} = \frac{25}{\pi} \]

### Step 5: Solve for the side length \( a \)
To find the side length \( a \), we rearrange the equation:
\[ a = \frac{25}{\pi\sqrt{2}} \]

### Final Answer
The side of the square inscribed in the circle is:
\[ a = \frac{25}{\pi\sqrt{2}} \text{ cm} \]

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