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

All rectangles of a given area will have the same perimeter,.Therefore knowing both will not help you to determine the dimensions.

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It is not true that all rectangles of a given area will have the same perimeter. To prove this to yourself, imagine a rectangle measuring 2 x 2, with a perimeter of 8, and area of 4. Then compare to a rectangle measuring 1 x 4, with a perimeter of 10, but with the same 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 perimeter 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) has
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 change the length to a positive number because when measuring length, positive or negative just depends on the direction)

You get 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 was 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

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