An online calculator to calculate the length and width (dimensions) of a rectangle given its area A and length of diagonal L. If x and y are the length and width of a rectangle, the area A and the length L of the diagonal may be expressed in terms of x and y as follows:
Area = A = x y
length of diagonal = L = √ (x^{ 2} + y^{ 2})
The calculator in this page solves the above equations in two variables and display the length and width.
solve A = x y for y to obtain y = A / x
Square both sides of the equation L = √ (x^{ 2} + y^{ 2}) to obtain L^{2} = x^{ 2} + y^{ 2}
Substitute y by A / x in the equation L^{2} = x^{ 2} + y^{ 2} to obtain
L^{ 2} = x^{ 2} + (A / x )^{ 2}
Multipliy all terms of the above equation by x^{ 2}, simplify and write it in standard for to obtain
x^{ 4}  L^{ 2}x^{ 2} + A^{ 2} = 0
Discriminant of above equation
Δ = (L^{2})^{2}  4(1) A^{ 2}
The problem of finding x and y has a solution if Δ ≥ 0
L^{4}  4 A^{ 2} ≥ 0
The condition for the rectangle to exist is: L ≥ √(2 A)
How to use the calculator
Enter the area A and the length L of the diagonal of the rectangle to solve, as positive real numbers, and press "calculate". The outputs are the dimensions, length x and width y, of the rectangle.
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