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Let's break this down:

We are given a square with a side length of 2, and two circles inscribed inside, each touching a diagonal of the square. The problem asks us to find the ratio of the **area of the circles** to the **area of the square**, in exact terms.

### Step 1: Area of the Square
The side length of the square is given as 2 units. The area \( A_{\text{square}} \) of the square is calculated by squaring the side length:
\[
A_{\text{square}} = 2^2 = 4 \text{ square units.}
\]

### Step 2: Diameter and Radius of Each Circle
Since the two circles are inscribed in such a way that their diameters are aligned along the diagonal of the square, the diagonal of the square is also the sum of the diameters of the two circles.

- The diagonal \( d \) of the square can be found using the Pythagorean theorem:
\[
d = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}.
\]

This diagonal represents the sum of the diameters of the two circles. Let the radius of each circle be \( r \), so the diameter of one circle is \( 2r \). Since there are two circles:
\[
2r + 2r = 2\sqrt{2} \quad \Rightarrow \quad 4r = 2\sqrt{2} \quad \Rightarrow \quad r = \frac{\sqrt{2}}{2}.
\]

### Step 3: Area of the Circles
The area \( A_{\text{circle}} \) of a circle is given by the formula:
\[
A_{\text{circle}} = \pi r^2.
\]
For each circle, substituting \( r = \frac{\sqrt{2}}{2} \):
\[
A_{\text{circle}} = \pi \left(\frac{\sqrt{2}}{2}\right)^2 = \pi \cdot \frac{2}{4} = \frac{\pi}{2}.
\]
Since there are two circles, the total area of the circles is:
\[
A_{\text{circles}} = 2 \times \frac{\pi}{2} = \pi \text{ square units.}
\]

### Step 4: Ratio of the Areas
Now, we can compute the ratio of the area of the circles to the area of the square:
\[
\frac{\text{Area of the circles}}{\text{Area of the square}} = \frac{\pi}{4}.
\]

### Final Answer:
The exact answer for the ratio of the area of the circles to the area of the square is:
\[
\boxed{\frac{\pi}{4}}.
\]

Would you like more details or have any questions?

Here are some related questions to deepen your understanding:
1. What would the ratio be if the side of the square were doubled?
2. How would the problem change if the square had side length 4?
3. What is the perimeter of the square in the given problem?
4. Can you compute the total length of the diagonal in a square of side 3?
5. How would the ratio change if there were three circles instead of two?

**Tip:** Understanding how shapes interact with diagonals can simplify many geometric problems, especially when dealing with inscribed circles or polygons.

Two circles are inscribed in a square of side 2 as shown below. What is the ratio of the area of the circles to the area of the square? Give the exact answer and simplify as far as possible.

Learn how to calculate the ratio of the area of two inscribed circles to the area of a square with side 2. This geometry problem explores key concepts including the Pythagorean Theorem, area of circles, and ratios. Suitable for students in Grades 9-12.

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