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 determine the length of each side of the octagon after the corners of the square are cut off.
The corners of the square are cut off to form an octagon, which means that each of the eight angles in the resulting shape is equal. In other words, each angle measures 45 degrees.
Since the square has sides of 2 meters, the diagonal across the square can be found using the Pythagorean theorem. The diagonal is equal to the square root of the sum of the squares of the two sides.
Using the formula: diagonal = √(2^2 + 2^2) = √8 = 2.83 meters (rounded to two decimal places).
In the resulting octagon, this diagonal becomes the side length of the octagon. Therefore, the length of each side of the resulting octagon is 2.83 meters.
To find the length of the side of the resulting octagon, we can start by considering the original square and the regular octagon formed by cutting off the corners.
Let's analyze the situation step by step:
1. Start with a square with sides of length 2 meters.
2. Cutting off the corners creates four isosceles right triangles at each corner of the square. In each triangle, the two legs are congruent and have a length equal to the side length of the original square (2 meters).
3. The hypotenuse of each of these right triangles is the side length of the resulting octagon.
4. To calculate the length of the hypotenuse, we can use the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the two legs of the right triangle have a length of 2 meters each. Let's use the formula:
c^2 = a^2 + b^2
Where c is the hypotenuse and a, b are the legs.
c^2 = 2^2 + 2^2
c^2 = 4 + 4
c^2 = 8
To find the value of c, we take the square root of both sides:
c = √8
Simplifying the square root, we have:
c ≈ 2.828 meters
So, the length of each side of the resulting octagon is approximately 2.828 meters.
Without loss of generality, assume a square of side 1. Now, if we cut off a piece of length x from each end of the side, then the diagonal of the cut corner will be x√2.
To make things come out right, the two cuts and the remaining middle piece add up to 1:
1 - 2x = x√2
1 = (2+√2)x
x = 1/(2+√2) = 1 - 1/√2 = 0.2928
Check:
1 - 2x = .4142
.2928√2 = .4142
So, I've solved for x. What is the side length?
Let x be the side of square and y be the side of octagon their for the relation between y and x is:
y=x/1+2^1/2
Theirfore y=.828