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, let's break down the problem and solve it step by step.
Step 1: Determine the original side length of the square.
Given that the original square has side length 2m, we'll use this value to calculate the length of the octagon's side.
Step 2: Calculate the length of the octagon's side using the side length of the square.
When you cut off the corners of the square, each removed triangle will be an isosceles right triangle because all the sides of the square are equal.
Step 3: Determine the side length of the triangle formed.
To calculate the length of the triangle's side, we need to find the hypotenuse of the triangle, which is the same as the side length of the octagon.
Step 4: Use Pythagorean theorem to find the hypotenuse.
In an isosceles right triangle, if the legs have length a, then the hypotenuse has length sqrt(2) * a. Therefore, we can calculate the length of the hypotenuse (side of the octagon) using the formula:
Length of hypotenuse = sqrt(2) * Length of triangle's leg
Step 5: Calculate the length of the octagon's side.
Since the triangle's leg is half of the original square's side (half of 2m), we can substitute it into the formula:
Length of hypotenuse = sqrt(2) * (Length of square's side / 2)
Plugging in the values, we get:
Length of hypotenuse = sqrt(2) * (2m / 2) = sqrt(2) * m = m * sqrt(2)
Therefore, the length of the side of the resulting octagon is m * sqrt(2), where m is the original side length of the square. In this case, since the original side length is given as 2m, the length of the resulting octagon's side is 2m * sqrt(2).