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To solve the problem, we need to determine the greatest possible diameter of the round post that can be cut from the square piece of wood, given that the diagonal of the square is 20 cm.

### Step-by-Step Solution:

1. **Understanding the problem**:  
   The diagonal of the square is given as 20 cm. We need to find the diameter of the circle that fits inside this square. The diameter of the circle will be equal to the side length of the square.

2. **Relating the diagonal to the side of the square**:  
   The relationship between the side length \( s \) of a square and its diagonal \( d \) is given by the formula:
   \[
   d = s\sqrt{2}
   \]
   Here, \( d = 20 \) cm.

3. **Calculate the side length of the square**:
   \[
   s = \frac{d}{\sqrt{2}} = \frac{20}{\sqrt{2}} = \frac{20 \times \sqrt{2}}{2} = 10\sqrt{2} \text{ cm}
   \]
   Simplifying further:
   \[
   s \approx 10 \times 1.414 = 14.14 \text{ cm}
   \]

4. **Diameter of the circle**:
   Since the circle fits perfectly within the square, the diameter of the circle will be equal to the side length of the square:
   \[
   \text{Diameter of the circle} = 14.14 \text{ cm}
   \]

### Final Answer:
The greatest possible diameter of the round post is approximately **14.14 cm**.

Do you have any questions or would you like further details?

Here are some related questions for further exploration:
1. What is the area of the circle that can be cut from the square?
2. If the diagonal of the square was doubled, what would the new diameter of the post be?
3. What is the perimeter of the square given the diagonal is 20 cm?
4. How would the answer change if the square's diagonal was only 10 cm?
5. Can you calculate the radius of the circle given the diameter?

**Tip:** When dealing with squares and circles, remember that the diameter of the inscribed circle is always equal to the side length of the square.

A round post is cut from a square piece of wood with a diagonal of 20 cm. What is the greatest possible diameter of the post?

Learn how to calculate the greatest possible diameter of a round post that can be cut from a square piece of wood with a diagonal of 20 cm. This problem involves key geometric concepts and formulas, including the relationship between a square's diagonal and its side length.

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