To calculate the area of the smooth-edged square table, we need to consider the specific geometry of the table, especially the smooth edges and the curve.

Given:

• The edge length of the square when straightened: $4 \, \text{cm}$
• The edge curve: from $0^\circ$ to $90^\circ$
• The space between the edges: $7 \, \text{cm}$

Assumptions and Interpretation:

1. Edge Length Interpretation: The edge length of $4 \, \text{cm}$ is the length of each side of the square if the edge were to be straightened out.
2. Curve of the Edge: Since the edge is curved from $0^\circ$ to $90^\circ$, it suggests that the table's sides are not straight but instead curved, possibly in the form of a quarter-circle.
3. Space Between the Edges: The space between the curved edges could imply the distance between the edges at their farthest points (possibly the diagonal).

Step 1: Calculate the Effective Length of One Side

For simplicity, assume the curve takes the form of a quarter-circle with a radius equal to half of the straightened edge length, $r = \frac{4}{2} = 2 \, \text{cm}$.

Step 2: Understanding the Curve

• A quarter-circle curve with a radius $r = 2 \, \text{cm}$ will have an arc length equal to $2 \, \text{cm}$.
• Each side of the square, when considering the curve, effectively curves outward, contributing to the perimeter and altering the area calculation.

Step 3: Calculate the Area

Given the above, we need to approximate the area of the smooth-edged square. However, the space between the edges being $7 \, \text{cm}$ suggests the table is slightly rounded with a significant diagonal stretch.

The area $A$ of the table can be approximated using the formula for the area of a square:

$A = s^2$

Where $s$ is the effective side length of the square. Given the complexity of the curvature and its effect, we approximate $s$ as being slightly more than $4 \, \text{cm}$.

However, if the edges were straight, the maximum diagonal would be:

$\text{Diagonal} = \sqrt{2} \times s$

For the given distance between the edges $7 \, \text{cm}$, let $s$ be approximately $5 \, \text{cm}$ to account for curvature.

$A \approx 5^2 = 25 \, \text{cm}^2$

Approximate Area:

Thus, the area of the smooth-edged square table is approximately 25 square centimeters.

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Would you like more details, or do you have any questions? Here are some related questions you might find interesting:

1. How do different curvatures of edges affect the area of shapes?
2. What are other methods to calculate the area of curved shapes?
3. How do you calculate the perimeter of a shape with curved edges?
4. What is the difference between effective and actual edge length in geometry?
5. Can we use calculus to find the exact area of irregular shapes?

Tip: When dealing with curves, try using approximations or geometric properties like arc length to simplify calculations.