Question

What are the side lengths of the rectangle area 40 perimeter 26

Answer 1

Recall the formula for finding the area of a rectangle:

           Area = Length × Width

Recall the formula for finding the perimeter of a rectangle:

         Perimeter = 2 ( Length + Width )

Given in your problem:
        Area = 40 sq. units
        Perimeter = 26 units

Required to solve for:
        Length (L) and width (W)

• First, substitute the given to the formula:

        Area = Length x Width
          40 = L × W      ⇒   equation number 1

       Perimeter = 2 ( Length + Width )
            26 = 2 ( L + W )     ⇒   equation number 2


• Simplifying equation number 2,
           13 = L + W

 • Rearranging the equation, 

         L = 13 - W   ⇒   equation 3

Substituting equation 3 from equation 1:

      ( equation 1 )               40 = (L)(W)
      ( equation 3 )                L = 13 - W
                                      40 = (13 - W) (W)
                                      40 = 13W - W² 
      ( regrouping )          W² - 13W + 40 = 0
      ( factoring )             (W - 8) (W - 5) = 0
                                     W - 8 = 0    ;     W - 5 = 0
                                        W = 8      ;        W = 5
   
Therefore, there are 2 possible values for the width of the rectangle. It can be 8 units or 5 units.

• Now to solve for the length of the rectangle, substitute the two values of width to equation 3.

      (equation 3)        L = 13 - W
                       for W = 8    ⇒   L = 13 - 8
                                                   L = 5 units
                   
                       for W = 5    ⇒  L = 13 - 5
                                                   L = 8 units