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 L2 = x 2 + y 2
Substitute y by (P - 2 x) / 2 in the equation L2 = 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 L2 - 16 P 2
For the problem to have a solution, we must have the condition Δ &ge 0 satisfied.
128 L2 - 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.