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In this problem, we are given a square that fits exactly inside a circle, with all of its vertices on the circumference of the circle. The square has sides of length \( x \) cm, and the area of the circle is \( 56 \, \text{cm}^2 \).

Let's break this down:

### Step 1: Find the radius of the circle
The formula for the area of a circle is:
\[
A = \pi r^2
\]
We are given that the area of the circle is \( 56 \, \text{cm}^2 \). So, we can solve for \( r \) (the radius):
\[
56 = \pi r^2
\]
\[
r^2 = \frac{56}{\pi}
\]
\[
r = \sqrt{\frac{56}{\pi}}
\]
Calculating this will give us the radius.

### Step 2: Relate the radius of the circle to the side length of the square
Since the square fits inside the circle with its vertices touching the circumference, the diagonal of the square is equal to the diameter of the circle.

The relationship between the side length \( x \) of the square and its diagonal \( d \) is given by:
\[
d = x\sqrt{2}
\]
But since the diagonal \( d \) is also equal to \( 2r \), we have:
\[
x\sqrt{2} = 2r
\]
Thus, we can solve for \( x \):
\[
x = \frac{2r}{\sqrt{2}} = r\sqrt{2}
\]

### Step 3: Final calculation
Now that we have the value of \( r \) from Step 1, we can substitute it into the equation \( x = r\sqrt{2} \) to find \( x \).

Let me calculate the exact value for you.The side length of the square, \( x \), is approximately \( 5.971 \, \text{cm} \) when rounded to three significant figures.

Would you like more details or have any further questions?

Here are five related questions to deepen your understanding:
1. How would the problem change if the area of the circle were doubled?
2. Can you derive a general formula for the side length of a square inside a circle of any area?
3. What would happen if the square were rotated inside the circle?
4. How would you calculate the area of the square in this case?
5. How can you apply this knowledge to other polygons inscribed in circles?

**Tip**: In geometry, it's often useful to relate the properties of inscribed shapes (like squares) to their circumscribing shapes (like circles) using diagonal and radius relationships.

A square fits exactly inside a circle with each of its vertices being on the circumference of the circle. The square has sides of length x cm. The area of the circle is 56 cm². Work out the value of x. Give your answer correct to 3 significant figures.

Learn how to calculate the side length of a square that fits perfectly inside a circle with a known area of 56 cm². This problem involves understanding the relationship between the diameter of the circle and the diagonal of the square and applying properties of squares and circles to find the solution.

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