A regular octagon is formed by cutting off the corners of a square. If one side of the square is n cm, find the total area removed in square cm.
To find the total area removed, we need to determine the area of the octagon.
First, let's calculate the area of the square. Since one side of the square is n cm, the area of the square is n * n = n^2 square cm.
To find the area of the octagon, we need to calculate the area of each triangle that is removed. Since there are 8 triangles, we will calculate the area of one triangle and multiply it by 8.
To find the area of a triangle, we need the length of the base and the height. Since the base of each triangle is a side of the square, it is equal to n cm.
Now, let's find the height of the triangle. If we draw a diagonal from one corner of the square to the opposite corner, it divides the square into two congruent right triangles. The diagonal is also the hypotenuse of the triangle.
Using the Pythagorean theorem, we can calculate the height of the triangle:
height = √(hypotenuse^2 - base^2)
= √(n^2 + n^2) [since the two legs are equal sides of the square]
= √(2n^2)
= √2 * n
Now that we have the length of the base and the height of the triangle, we can calculate the area of one triangle using the formula: area = 0.5 * base * height. Substituting the values, we get:
area = 0.5 * n * (√2 * n)
= 0.5 * n * √2 * n
= √2 * 0.5 * n^2
= √2 * (n^2 / 2)
= (√2 / 2) * n^2
Since we have 8 triangles, the total area removed is:
total area removed = 8 * (√2 / 2) * n^2
= 4√2 * n^2
Therefore, the total area removed is 4√2 * n^2 square cm.
If the length of the corners cut off is s, the diagonal length is s√2.
The area of each triangle is thus 1/2 (2s^2) = s^2
Now, we know that 2s + s√2 = n, so
s = n/(1+√2)
The area cut off is thus 4n^2/(3+2√2) = 4(3-√8)n^2