# Exact Value for a Diagonal of a Square

• Dec 19th 2012, 07:57 AM
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.

• Dec 19th 2012, 10:07 AM
Plato
Re: Exact Value for a Diagonal of a Square
Quote:

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

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

• Dec 19th 2012, 12:06 PM
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.
• Dec 19th 2012, 12:30 PM
emakarov
Re: Exact Value for a Diagonal of a Square
Indeed, $\displaystyle 2\sqrt{2}+2\sqrt{6}=4\sqrt{2+\sqrt{3}}$, which you can verify by squaring both sides.
• Dec 19th 2012, 01:43 PM
bjhopper
Re: Exact Value for a Diagonal of a Square
key concept 6^1/2 = 3^1/2 * 2^1/2
• Dec 19th 2012, 04:04 PM
cathectio
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).
• Dec 19th 2012, 04:08 PM
cathectio
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.
• Dec 19th 2012, 05:45 PM