# Rectangle Perimeter and Diagonal Calculator

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*to obtain

^{ 2}+ y^{ 2})*L*

^{2}= x^{ 2}+ y^{ 2}Substitute

*y*by

*(P - 2 x) / 2*in the equation

*L*to obtain

^{2}= x^{ 2}+ y^{ 2}*L*

^{ 2}= x^{ 2}+ ((P - 2 x) / 2)^{ 2}*L*/ 4

^{ 2}= x^{ 2}+ (P - 2 x)^{ 2}Multipliy all terms of the above equation by

*4*, simplify and write it in standard for to obtain

*8x*- 4L

^{ 2}- 4 P x + P^{ 2}^{ 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 ≥ 0 satisfied.

128 L

^{2}- 16 P

^{ 2}≥ 0

*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 Rectangle Perimeter and Diagonal 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.

## More References and Links

Rectangle Area and Diagonal Calculator

Online Geometry Calculators and Solvers.