To determine the side length of a regular octagon that fits inside a 16-inch square, we can break the problem down geometrically. The octagon is inscribed within the square, and we need to relate the side length of the octagon to the dimensions of the square.

Step-by-Step Approach:

1. Square Dimensions:
   The side length of the square is $16$ inches.

2. Relationship Between the Octagon and the Square:
   A regular octagon can be inscribed in a square by cutting off the four corners of the square, forming right-angled triangles at each corner. The side length of the octagon is related to the side length of the square.

3. Formula for the Side Length of the Octagon:
   If the side length of the square is $s$, then the side length of the octagon $a$ can be found using the following formula: $a = \frac{s}{1 + \sqrt{2}}$ where $s = 16$ inches.

4. Calculation: $a = \frac{16}{1 + \sqrt{2}}$

   First, calculate $1 + \sqrt{2}$: $1 + \sqrt{2} \approx 1 + 1.414 = 2.414$

   Now, divide the side length of the square by this value: $a = \frac{16}{2.414} \approx 6.63 \text{ inches}$

Thus, the length of each side of the octagon is approximately 6.63 inches.

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Would you like more details on the geometric approach or other related concepts?

Here are five related questions:

1. How do we derive the formula for the side length of a regular octagon inscribed in a square?
2. What is the area of the regular octagon formed inside the 16-inch square?
3. How does the side length of a regular polygon relate to its circumradius or inradius?
4. What would the dimensions be if the square's side length were different, say 20 inches?
5. Can this approach be applied to other regular polygons, such as a regular hexagon?

Tip: For regular polygons inscribed in a circle or square, the side length can often be related using trigonometric or geometric formulas based on angles and symmetry!