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 can use the Pythagorean theorem.

First, let's consider one of the triangles formed by cutting off a corner of the square. Each triangle is isosceles, with two sides of length 2m and an angle of 45 degrees at the vertex.

Now, draw the diagonal from the vertex of the triangle to the opposite vertex of the square. This diagonal divides the triangle into two right triangles. Since each angle of the square is 90 degrees, these right triangles are congruent.

By the Pythagorean theorem, we can find the length of the diagonal.

Let's call the length of the diagonal d. We can set up the equation:

(d/2)^2 + (d/2)^2 = 2^2

Simplifying it, we have:

(1/2) * d^2 + (1/2) * d^2 = 4

Combining like terms, we get:

d^2 = 8

Taking the square root of both sides, we find:

d = √8 = 2√2

Since the diagonal divides the octagon into two congruent right triangles, the length of one side of the octagon is half the length of the diagonal.

Therefore, the length of one side of the resulting octagon is:

2√2 / 2 = √2

So, the length of the side of the resulting octagon is √2 meters.

Let the side length be a.

a + 2*a*(sqrt2)/2 = 2
a = 2/(2.414) = 0.8285 m

See http://mathcentral.uregina.ca/QQ/database/QQ.09.02/rich1.html
for a proof