Length and Width of Rectangle - Calculator

Online calculator to calculate the dimensions (length and width) of a rectangle given the area A and perimeter P of the rectangle.

Formula for Area and Perimeter of Rectangle

The formulas for the perimeter P and the area A of the rectangle are used to write equations as follows: \[ P = 2 \times L + 2 \times W \] \[ A = L \times W \] Then these equations are solved for L and W which are the length and width of the rectangle.

How to use the calculator

Enter the perimeter P and area A as positive real numbers and press "Calculate". The outputs are the width, length and diagonal of the rectangle. There are conditions under which this problem has a solution (see formulation of problem below).

Formulation of Problem

Let P be the perimeter of a rectangle and A its area. Let W and L be, respectively, the width and length of the rectangle. Find W and L in terms of P and A.

solution

\[ P = 2 \times W + 2 \times L \quad (1) \] and \[ A = W \times L \quad (2) \]
solve equation (1) for W: \[ W = P / 2 - L \] substitute \( W \) by \( P / 2 - L \) in \( A = W \times L \) to obtain \[ A = L \times (P / 2 - L) \] Rewrite as a standard quadratic equation in \( L \) \[ L^2 - L (P / 2) + A = 0 \] Solve the above equation for \( L \) and find \( W \) using \( W = P / 2 - L \).

Practice: Find the discriminant of the above quadratic equation and find the condition on P and A under which the above problem has no solution.