Find the square of 39 by using the diagonal method.

Answer

To find the square of 39 using the diagonal method, we will follow these steps:

### Step-by-Step Solution:

1. **Draw a Square and Divide it into Four Sub-squares**:
- Start by drawing a large square and then divide it into four equal smaller squares. Label these squares as follows:
- Top-left: A
- Top-right: B
- Bottom-left: C
- Bottom-right: D

**Hint**: Remember that each sub-square represents a part of the multiplication process.

2. **Write the Number 39 in the Squares**:
- Write the number 39 along the left side (in squares A and C) and again along the top side (in squares A and B). It should look like this:
```
3 | 9
------
3 |
------
9 |
```

**Hint**: The number 39 can be broken down into its digits: 3 and 9.

3. **Multiply Each Digit**:
- Now, multiply the digits corresponding to each sub-square:
- A (3 * 3) = 9
- B (3 * 9) = 27
- C (9 * 3) = 27
- D (9 * 9) = 81

**Hint**: Write the products in their respective sub-squares and remember to carry over if necessary.

4. **Fill in the Sub-squares**:
- Fill in the results from the multiplications:
```
9 | 27
------
27 | 81
```

**Hint**: Make sure to keep track of the digits and their positions.

5. **Add the Diagonal Values**:
- Now, we will add the values along the diagonals:
- From top-left to bottom-right: 9 (from A) + 27 (from B) + 81 (from D) = 117
- From top-right to bottom-left: 27 (from B) + 27 (from C) = 54

**Hint**: When adding, remember to separate the tens and units to carry over if needed.

6. **Combine the Results**:
- Combine the results from the diagonal sums:
- For 117, write down 7 and carry over 11.
- For 54, add the carry over 11: 5 + 11 = 16. Write down 6 and carry over 1.

**Hint**: Keep track of the carry-over values carefully.

7. **Final Result**:
- Now, write down the final result from the diagonals:
- The final digits are: 1 (from the carry) 5 2 1, which gives us 1521.

**Hint**: The final answer represents the square of the original number.

### Conclusion:
The square of 39 is **1521**.