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 follow these steps:
1. Determine the length of the diagonal of the square:
- The square has sides of 2 meters each.
- The diagonal is the hypotenuse of a right triangle with sides of 2 meters each.
- By using the Pythagorean theorem (a^2 + b^2 = c^2), we can find the length of the diagonal (c).
- In this case, a = b = 2 meters.
- Thus, c^2 = 2^2 + 2^2 = 4 + 4 = 8.
- Taking the square root of both sides, we find that c = sqrt(8) ≈ 2.83 meters.
2. Determine the length of one side of the resulting octagon:
- When the corners of a square are cut off to form a regular octagon, the octagon is inscribed within the diagonal of the square.
- Since the diagonal of the square is also the diameter of the inscribed octagon, each side of the octagon is equal to the radius of the circle encompassing the octagon.
- Thus, the length of one side of the resulting octagon is equal to half the length of the diagonal.
- In this case, the length of one side of the octagon is 2.83/2 = 1.415 meters.
Therefore, the length of the side of the resulting octagon is approximately 1.415 meters.