On a 90 degree square,what ratio is the 45degree line longer than the lines that make up the square?

I think it is around 4/5 longer,but this is not hundred percent true,as the the dimensions increase the inaacuracie increases.I would like to know what is the exact calculation for this,if it is possible.Answers much appreciated.On a 90 degree square,what ratio is the 45degree line longer than the lines that make up the square?
Square root of 2





Edit: I'll explain a^2 + b^2 =c^2, so c = sqrt(a^2+b^2)





for a square a=b and we'll say they equal 1, so c = sqrt(1+1)=sqrt(2)On a 90 degree square,what ratio is the 45degree line longer than the lines that make up the square?
Let the sides of the square be length x.





The the length of the diagonal is d is given by Pythagoras' theorem.





d^2 = x^2 + x^2 = 2x^2





d = sqrt (2x^2)





ratio of d to x is d:x = sqrt(2x^2):x





when x = 1, d:x = sqrt (2*1^2): 1





= sqrt(2):1





when x = 2, d:x = sqrt(2*2^2) : 2





= sqrt(2*2*2):2





= 2 sqrt(2):2





= sqrt(2):1





So you can see the ratio is independent of x





Answer: sqrt(2):1
it's sqrt2 longer.





let's assume that the sides of square is 1, using the pythagorean theorem, we need to find the hypotenuse, so 1^2 + 1^2 = c^2





1 + 1 = c^2,





2 = c^2





square root of 2 = c