If a regular octagon is inscribed in a square

Hi...

A REGULAR octagon is an octagon with EQUAL sides..

but in your sketch, that is not the case.
4 sides are equal to SIDE of square and other 4 sides are equal to DIAGONAL of square..
that is why the answer is wrong.

But in the solution, you have written
\((2a+a\sqrt{2})^2=2a^2(1+\sqrt{2})^2\)

Hi

Side of Regular octagon will be in the middle of the side of square as also shown in the sketch in initial post. let the portion be 'a' on either end.
Each side of the square will be a+(side of octagon)+a.

Thus, each vertices of square will have small portions 'a' as the two side and third side will be the hypotenuse, which is nothing but the side of Octagon.

As each vertices of a square makes a right angle, the corner triangle formed will be a:a:a\(\sqrt{2}\), an isosceles right angled triangle.

No, it does not. Although a sketch would be easier to explain, but I will try through words.

Let us assume that it does trisect.
so you have each side of square is 3x, and is trisected as x+x+x. Also, every side of octagon too is x.
If this is true, take the triangles on any corner of square. The triangle will have two sides x each on its neighboring sides and the third side will also be x, given by the octagonal side. So, the triangle is x-x-x or an equilateral triangle. This means all angle of the triangle including the corner angle is 60. But the square has to have all angles as 90.
Thus, our assumption that each side of square is trisected is wrong.

Ill add a video or a sketch if this doesnt help.

hi chetan2u,
thank you for pointing out my error.
gracie

Can you explain - how you got the Side of the square after assuming that the side of the Octagon is a√2?

Also, are all sides of Octagons always equal?

Can you please explain the highlighted part? I think it is wrong. The area of square should be \(a^2(2+\sqrt{2})^2\)

Both are the same

\((2a+a\sqrt{2})^2=(a(2+\sqrt{2}))^2\)

Take out a

\(a^2(2+\sqrt{2)}^2\)

\((2a+a\sqrt{2})^2=(a(2+\sqrt{2}))^2\)

Take out a

\(a^2(2+\sqrt{2)}^2\)

Take out \(\sqrt{2}\) to get what you are asking about.

\(a^2(\sqrt{2}*\sqrt{2}+\sqrt{2})^2\)
\(a^2(\sqrt{2})^2(\sqrt{2}+1)^2\)
\(2a^2(\sqrt{2}+1)^2\)

how you considered those triangles formed to be right isosceles triangle? please explain

But every octagon side trisects every square sides right? so if one of the side of octagon is a square root 2 so the other two portions as shown in the sketch cant they be a square root 2?

I hope this helps. Please remember, we have to keep the sides of the octagon the same as I have considered x here, but the sides of squares will be obtained based on the sides of the octagon.