The corners of the square, 2m. sides, are cut off to form a regular octagon. What is the length of the side of the resulting octagon?

To find the length of the side of the resulting octagon, we need to understand the relationship between a square and an octagon that is formed by cutting off its corners.

Here's how you can solve this problem step by step:

1. Visualize the square: Start by imagining a square with 2m sides. This square has four corners.

2. Cut off the corners: Cut off the corners of the square uniformly to form an octagon. This means that each cut removes the same amount from each corner.

3. Identify the resulting shape: The shape formed by cutting off the corners of a square uniformly is called a regular octagon. A regular octagon is an eight-sided polygon with all sides of equal length.

4. Determine the length of the octagon's side: To calculate the length of the octagon's side, we can use a formula. The formula for finding the length of the side of a regular octagon is:

Side length of the octagon = Square side length - 2 * (Length of the cut-off portion)

In this case, the square side length is 2m. Since the corners are cut off uniformly, the length of the cut-off portion is the same for each corner. We need to determine this side length, and then we can substitute it into the formula to find the length of the side of the octagon.

5. Find the length of the cut-off portion: To find the length of the cut-off portion, we can draw a right triangle inside the square, connecting the midpoint of one side with the corner. This right triangle has a hypotenuse equal to the side length of the square and legs of equal length.

Using the Pythagorean theorem, we can determine the length of the legs of the right triangle:

Leg length = (Square side length) / √2

Substituting the value of the square side length, we get:

Leg length = 2m / √2

Simplifying further, we have:

Leg length = 2m / (√2) * (√2) / (√2)

Leg length = (2m * √2) / 2

Leg length = √2 * m

So, the length of the cut-off portion of each corner is √2 * m.

6. Substitute the values into the formula: Now that we know the length of the cut-off portion, we can substitute it into the formula to find the side length of the octagon:

Side length of the octagon = 2m - 2 * (√2 * m)

Simplifying the expression, we get:

Side length of the octagon = 2m - 2√2 * m

Factoring out m, we have:

Side length of the octagon = m * (2 - 2√2)

This is the final result for the length of the side of the resulting octagon after cutting off the corners of the square with 2m sides.