How Do You Find The Dimensions Of A Rectangle If You Know The Perimeter And The Area?


Answer:
All rectangles of a given area will have the same bounds,.Therefore knowing both will not help you to determine the dimensions.
Comment: This is a question in my neice's math homework. It is within the chapter about Quadradic & Exponential Functions.
Answer: It is not true that all rectangles of a given area will hold the same perimeter.  To prove this to yourself, imagine a rectangle measure 2 x 2, with a perimeter of 8, and area of 4.  Then compare to a rectangle measure 1 x 4, with a perimeter of 10, but with impossible to tell apart area of 4.  To solve the problem of your question, you will need to do some algebra, as follows:
  Perimeter = P = 2 (Length) + 2(Width) = 2L + 2W
  Area = A = (Length)(Width) = LW
Rearrange the equation for edge to find W:
  P = 2L + 2W = 2( L + W)
  (P/2) = L + W
  W = (P/2) - L
If you substitute this value into the Area equation,
  A = LW = L( (P/2) - L) = (PL/2)-(L^2)
  A = -L^2 + (PL/2)
Or, rearranging into standard form,
  L^2 -(P/2)L+A=0
Now you can use the quadratic formula to solve for L.

Remember that for an equation in standard form,
  aX^2 + bX + c = 0

  x = (-b +/- sqrt(b^2 - 4ac)) / 2a

So our equation (where x is L) have
   a = 1
   b = (P/2)
   c = A

Which gives

   L = (-(P/2) +/- sqrt((P/2)^2 - 4A)) / 2

Now, given perimeter and area, you can solve for the dimensions.  As an example, P = 15,   A = 14

  L = (-(15/2) +/- sqrt((15/2)^2 - 4*14)) / 2

  L =  (-7.5 +/- .5) / 2  =  -8 / 2   OR   -7 / 2

  L = 4  OR 3.5
(You can adaptation the length to a positive number because when measuring length, positive or negative just depends on the direction)

You draw from 2 answers due to the way the quadratic formula works.  However, it makes sense, because you would get two different answers depending on whether the Length or Width of your rectangle be the long dimension.  So, choosing L=4, the dimensions of the recangle would be 4 x 3.5 .
As a check,
  A = LW = (4)(3.5) = 14
  p = 2L + 2W = 2(4) + 2(3.5) = 8 + 7 = 15
To find the area and perimeter of a rectangle you need to know formulas for respectively:
Area of a rectangle=L*W
Perimeter of a rectangle=2(L+W)
So to find either the area or the perimeter of a rectangle you have need of to know to things.
The value of:
The rectangles length
The rectangles width
To illustrate lets suppose that all along the rectangle is 2 and the width is 3. Hence:
length= 2
width= 3
Area=2*3
Area=6
Perimeter=2(2+3)
Perimeter=2(5)
Perimeter=10


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