The side of a square is 10 cm long. What is the circumference of the circle when a circle is circumscribed around the square?

To find the circumference of a circle circumscribed around a square, we need to first determine the radius of the circle. A circle circumscribed around a square will have its diameter equal to the diagonal of the square.

### Steps:

1. **Find the diagonal of the square**:  
   The diagonal \( d \) of a square can be calculated using the Pythagorean theorem, since the diagonal forms the hypotenuse of a right triangle where the two sides are the side lengths of the square.  
   For a square with side length \( s = 10 \, \text{cm} \):

   \[
   d = \sqrt{s^2 + s^2} = \sqrt{10^2 + 10^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} \, \text{cm}
   \]

2. **Determine the radius**:  
   The radius \( r \) of the circumscribed circle is half the diagonal:

   \[
   r = \frac{d}{2} = \frac{10\sqrt{2}}{2} = 5\sqrt{2} \, \text{cm}
   \]

3. **Find the circumference of the circle**:  
   The circumference \( C \) of a circle is given by the formula:

   \[
   C = 2\pi r
   \]

   Substituting the value of \( r = 5\sqrt{2} \):

   \[
   C = 2\pi \times 5\sqrt{2} = 10\pi\sqrt{2} \, \text{cm}
   \]

So, the circumference of the circumscribed circle is \( 10\pi\sqrt{2} \, \text{cm} \), or approximately **44.43 cm** when evaluated numerically.

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Do you want further details or have any questions about this?

Here are some related questions:
1. What is the area of the square in this scenario?
2. How would you calculate the area of the circumscribed circle?
3. What is the length of the diagonal of the square in decimal form?
4. Can you find the perimeter of the square and compare it with the circle’s circumference?
5. How would you compute the area difference between the square and the circle?

**Tip**: When dealing with a circumscribed circle around a square, always start by finding the diagonal of the square, as it becomes the diameter of the circle.

Find the circumference of a circle circumscribed around a square with a side length of 10 cm. This problem involves applying the Pythagorean theorem to find the diagonal of the square and using the formula for the circumference of a circle. Suitable for Grades 8-10 students.

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