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We are tasked with finding the area of each square on the checkerboard. Here's how we can approach the problem:

### Step 1: Analyze the Given Information
- The checkerboard has 64 congruent squares. Since the checkerboard is an 8x8 grid, the total number of squares is \(8 \times 8 = 64\).
- The diagonal length of the entire checkerboard is given as 40 cm.

### Step 2: Relate the Diagonal to the Checkerboard's Side Length
For an 8x8 checkerboard, the side length of the checkerboard consists of 8 squares. If the side length of one square is \( s \), the total side length of the checkerboard is \( 8s \).

The diagonal of the checkerboard forms a right triangle, with the two legs equal to the side length of the checkerboard. Using the Pythagorean theorem:
\[
d^2 = (8s)^2 + (8s)^2
\]
\[
d^2 = 2(8s)^2
\]
\[
d = 8s\sqrt{2}
\]
Given that the diagonal \( d = 40 \) cm, we substitute:
\[
40 = 8s\sqrt{2}
\]
Solve for \( s \):
\[
s = \frac{40}{8\sqrt{2}} = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}{2} \approx 3.535 \text{ cm}
\]

### Step 3: Find the Area of Each Square
The area \( A \) of each square is:
\[
A = s^2 = \left(\frac{5\sqrt{2}}{2}\right)^2 = \frac{50}{4} = 12.5 \text{ square cm}
\]

Thus, the area of each square on the checkerboard is \( 12.5 \, \text{cm}^2 \).

Would you like further details or have any questions?

### Relative questions:
1. How does the Pythagorean theorem apply to this scenario?
2. What would the diagonal of the checkerboard be if the area of each square was 16 cm²?
3. How does the diagonal of the entire checkerboard change if the number of squares increases?
4. Can you derive the formula for the diagonal of any checkerboard size?
5. How would this problem change if the diagonal length was measured in a different unit?

### Tip:
When solving geometric problems involving diagonals, the Pythagorean theorem is often useful, especially when the figure can be divided into right triangles.

A checkerboard has 64 congruent squares. Suppose a checkerboard has a diagonal length of 40 cm. Find the area of each square on the board.

Learn how to find the area of each square on a checkerboard with 64 squares and a diagonal of 40 cm. This geometry problem uses the Pythagorean theorem and algebraic manipulation to determine the side length and area of each square. Suitable for students in Grades 8-10.

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