MHB Find side length, cirumference and area of octagon

[Daugava]

Square-shaped piece of paper is intended to make a regular octagon through the cutting of the vertices of a square.
The length of the piece of paper is 29 cm.
How long triangle cathetus have to be cut off from the vertices of the square?
Calculate the octagonal side length, circumference, and area.
Sorry for my english, if it's not understandable I'll try to explain it better.. but how do I do this?
View attachment 8187

 

[Opalg]

Square-shaped piece of paper is intended to make a regular octagon through the cutting of the vertices of a square.
The length of the piece of paper is 29 cm.
How long triangle cathetus have to be cut off from the vertices of the square?
Calculate the octagonal side length, circumference, and area.
Sorry for my english, if it's not understandable I'll try to explain it better.. but how do I do this?

Hi Daugava, and welcome to MHB!

You want to cut four triangles from the square of paper. Suppose that the shorter sides of these triangles have length $x$ cm. Then the hypotenuse of the triangle will be $\sqrt2x$ cm. After cutting off the triangles, the sides of the square will have been shortened by $2x$ cm. You want all the sides of the resulting octagon to have the same length. That implies that $\sqrt2x = 29 - 2x$. Solve that equation to find the octagonal side length. It should then be easy to find the circumference of the octagon. For the area of the octagon, subtract the area of the four triangles from the area of the square.

 

[Daugava]

So the side length is 12 cm
a = 12 cm
Area is 2*(1+sqrt2)a^2 = 695,3 cm^2
Circumference is 8*a = 96 cm
Is this right?
But how do I find out how long the triangle cathetus/sides are that were cut off from the vertices of the square?
Is the cut off 8,5 cm?

 

[Opalg]

So the side length is 12 cm
a = 12 cm
Area is 2*(1+sqrt2)a^2 = 695,3 cm^2
Circumference is 8*a = 96 cm
Is this right?
But how do I find out how long the triangle cathetus/sides are that were cut off from the vertices of the square?
Is the cut off 8,5 cm?

Thank you for teaching me a new word in my own language! I did not know that a cathetus is one of the perpendicular sides of a right-angled triangle.

The side length is not precisely 12. In fact, it is $29(\sqrt2-1)$, which is approximately 12.012. The perimeter is approximately 96.0975.

The value for the cathetus comes from my previous comment above. If the cathetus is $x$ cm, then $\sqrt2x = 29 - 2x$. Solve that equation to get $x = \dfrac{29}{2+\sqrt2}.$