An online calculator to calculate the length and width (dimensions) of a rectangle given its perimeter P and length of diagonal L. If x and y are the length and width of a rectangle, the perimeter P and the length L of the diagonal may be expressed in terms of x and y as follows:
Perimeter = P = 2 x + 2 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 P = 2 x + 2 y for y to obtain y = (P  2 x) / 2
Square both sides of the equation L = √ (x^{ 2} + y^{ 2}) to obtain L^{2} = x^{ 2} + y^{ 2}
Substitute y by (P  2 x) / 2 in the equation L^{2} = x^{ 2} + y^{ 2} to obtain
L^{ 2} = x^{ 2} + ((P  2 x) / 2) ^{ 2}
L^{ 2} = x^{ 2} + (P  2 x)^{ 2} / 4
Multipliy all terms of the above equation by 4, simplify and write it in standard for to obtain
8x^{ 2}  4 P x + P^{ 2}  4L^{ 2} = 0
Discriminant of above equation
Δ = (4 P)^{2}  4(8)(P^{ 2}  4L^{ 2}) = 128 L^{2}  16 P^{ 2}
For the problem to have a solution, we must have the condition Δ &ge 0 satisfied.
128 L^{2}  16 P^{ 2} ≥ 0
The condition for the rectangle to exist is: L ≥ P / (2 √ 2)
Given P and L, once you solve the above equation for x, you also have to ckeck that y = (P  2 x) / 2 is positive.
How to use the calculator
Enter the perimeter P 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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