How Do You Calculate the Area of a Regular Octagon Cut from a Square?

[Idiotinabox]

Homework Statement

The question says that there is a square that has a side of length 1cm. It then says 4 equal triangles are cut, one off each corner. The resulting shape is a regular octagon. What is its area?

Homework Equations

I believe the sides of the triangles to both be (1-x)/2 assuming we let each side of the octagon be length x. This means that length x is represented by (triangle side length)^2 + (triangle side length)^2 = x^2.

The Attempt at a Solution

I essentially let the above equations equal and I get...
(0.5 - x^2/2)^2 + (0.5 - x^2/2)^2 = x^2
Using quadratic after simplifying the above...
X= rt2 - 1
From there I let that equal x again. I find the total area of all 4 triangles and get 1- area of four triangles which leaves me with 0.828cm^2

Is this correct? I'm not convinced it is actually correct. Any guidance?

 

[PeterO]

Homework Statement

The question says that there is a square that has a side of length 1cm. It then says 4 equal triangles are cut, one off each corner. The resulting shape is a regular octagon. What is its area?

Homework Equations

I believe the sides of the triangles to both be (1-x)/2 assuming we let each side of the octagon be length x. This means that length x is represented by (triangle side length)^2 + (triangle side length)^2 = x^2.

The Attempt at a Solution

I essentially let the above equations equal and I get...
(0.5 - x^2/2)^2 + (0.5 - x^2/2)^2 = x^2
Using quadratic after simplifying the above...
X= rt2 - 1
From there I let that equal x again. I find the total area of all 4 triangles and get 1- area of four triangles which leaves me with 0.828cm^2

Is this correct? I'm not convinced it is actually correct. Any guidance?

I got that answer calculating a different way, so it is probably correct.

 

[LCKurtz]

Homework Statement

The question says that there is a square that has a side of length 1cm. It then says 4 equal triangles are cut, one off each corner. The resulting shape is a regular octagon. What is its area?

Homework Equations

I believe the sides of the triangles to both be (1-x)/2 assuming we let each side of the octagon be length x. This means that length x is represented by (triangle side length)^2 + (triangle side length)^2 = x^2.

The Attempt at a Solution

I essentially let the above equations equal and I get...
(0.5 - x^2/2)^2 + (0.5 - x^2/2)^2 = x^2
Using quadratic after simplifying the above...
X= rt2 - 1
From there I let that equal x again. I find the total area of all 4 triangles and get 1- area of four triangles which leaves me with 0.828cm^2

That's correct, but why give a decimal approximation when you have an exact answer:$$
1-\frac 2 {(\sqrt 2 + 2)^2}$$

 

[Bohrok]

A better simplification would be 2(√2 - 1)