I don't even know what this question is asking?

Question

Question:

Find the dimensions of the triangle that must be cut from the corners of a square with side x to form a regular octagon.

Please show how or explain how this question is done.

Answer 1

An regular octagon is a polygon with 8 equal sides.

A square is a polygon with 4 equal sides.

If you take a square and cut off 4 corners with a diagonal cut (taking care to leave a part of the original sides), you will end up with an 8-sided figure.

The trick is to calculate where to make the cut so that the hypotenuse of the triangle you remove will be the same length as the portion of the original side that you will leave, to make sure that all 8 sides (and 8 angles) are equal.

The angle part tells you that the triangle you remove will have 45 degree angles at the hypotenuse.

Answer 2

The square starts off with a length of x.
You need to cut off four right triangles with legs y and hypotenuse y√2, such the when you cut off the triangles off the sides of the square, what you're left with is a length equal to the hypotenuses. There are two cuts on each side of the square, so you have:
x - 2y = y√2. Adding 2y,
x = 2y + y√2. Factoring y,
x = y(2 + √2). Dividing,
y = x / (2 + √2), or approximately x / 3.4142.
This will give a regular octagon. Since y is the leg of the right triangle, the dimensions are:
x / (2 + √2) high, x / (2 + √2) wide, with a hypotenuse of
(x√2) / (2 + √2).

Answer 3

OK, draw a square. Now draw a diagonal line from a point near the corner of one side to a spot on the adjacent side. See the little triangle you've formed? That's what they mean. If you do this on all four sides of the square you'll have an octagon in the center.

Now the question is how to do this to form a *regular* octagon. This means all sides of the octagon are the same length. If you draw these diagonals starting at 1/3 of the end of a side and going to 1/3 of the adjacent side, you'll get a regular octagon, with sides equal to 1/3 of the length of the side of the original square.

Answer 4

Like the question says, cut the corners off a square to make a shape with 8 equal sides. The correct answer does not involve 1/3rds!

Imagine a point at the center of your shape.

The shortest distance from the center to the edge is x/2.
The angle of each segment is 8/360=45 degrees
Call the octagon side that you want to find length y.
Split the triangle into two to find half the size of the side so
tan(45/2)=y/x (the 2s cancel out)
or y=x*tan(45/2)
so
y=0.414x
Find the other sides by pythagoras
z=sqrt((0.414^2)/2)=0.29x

To sum up, the triangle dimensions will be
0.29x on the equal sides and 0.414x on the long side.

Answer 5

draw a square
mark each side 1/3 and 2/3 along
cut the corners and you have an octogon

so the triangles to cut are isosceles of side = sqaure side/3

Answer 6

If you have a square piece of paper and cut a triagle off each corner, you will have an octagonal piece of paper. What proportion to the square is each triangle in order to have a uniform octagon?