# Thread: Exact Value for a Diagonal of a Square

1. ## Exact Value for a Diagonal of a Square

So, I understand the formula to achieve the diagonal of a square is a variation of Pythagoras' Theorem, which is the square root of 2 times the length squared.

My exact side length is 2 + 23. So I input this length in the formula and I get an answer of 42, however, in the book the answer is 4√2+√3, with the √3 being under the first radical.

I cannot see how they are getting to this answer, any help would be fantastic.

2. ## Re: Exact Value for a Diagonal of a Square

So, I understand the formula to achieve the diagonal of a square is a variation of Pythagoras' Theorem, which is the square root of 2 times the length squared.
My exact side length is 2 + 23. So I input this length in the formula and I get an answer of 42, however, in the book the answer is 4√2+√3, with the √3 being under the first radical.

There are several mistakes in this post.

If $d$ is the length of the diagonal of a square of side length $s$ then $d=s\sqrt{2}$

So if you have $s=2+2\sqrt{3}$ then $d=2\sqrt{2}+2\sqrt{6}$.

3. ## Re: Exact Value for a Diagonal of a Square

I see that d=s√2 is a simplified version of the formula I wrote. Still, I'm looking at the answer in the book and it is what I wrote. I suppose it could be a typo but it was also written this way in the answer book at school. I've heard my teacher say that the book sometimes answers questions in a strange way... I guess I will have to wait until tomorrow and ask my teacher.

Thanks again.

4. ## Re: Exact Value for a Diagonal of a Square

Indeed, $2\sqrt{2}+2\sqrt{6}=4\sqrt{2+\sqrt{3}}$, which you can verify by squaring both sides.

5. ## Re: Exact Value for a Diagonal of a Square

key concept 6^1/2 = 3^1/2 * 2^1/2

6. ## Re: Exact Value for a Diagonal of a Square

Blinkin is correct, as far as s/he goes, but 2sqrt(6) = 2sqrt(2*3), so we have then 2sqrt(2) + 2sqrt(2*3) = 2sqrt(2) + 2sqrt(2)*sqrt(3) = 2(2sqrt(2)) + sqrt(3) = 4sqrt(2) + sqrt(3).

7. ## Re: Exact Value for a Diagonal of a Square

It is of interest that in euclid's "elements", book i proposition i, if we allow the radius to equal 1 and duplicate the equilateral triangle vertically, we have the length ascertained of sqrt(3), which is the long hi to low opposite to opposite corner diagonal of a cube.

8. ## Re: Exact Value for a Diagonal of a Square

Originally Posted by cathectio
2sqrt(2) + 2sqrt(2*3) = 2sqrt(2) + 2sqrt(2)*sqrt(3) = 2(2sqrt(2)) + sqrt(3) = 4sqrt(2) + sqrt(3).
I'm not sure I understand this. Why does it become 2(2sqrt(2)) aren't we adding? And why is √3 still underneath the radical.