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

IS THIS EQUATION IS DIMENSIONALLY CORRECT?
P= L + W   where P is the perimeter, L   is the length, and W is the width.
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Times the area by the perimeter and that will give you the answer.
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Length of rectangle is 3 cm more than twice its width, area=35 cm2
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45
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To find the area and perimeter of a rectangle you need to know formulas for each:
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 need to know to things.
The value of:
The rectangles length
The rectangles width
To illustrate lets suppose that the length of 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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Width equal 7 feet less than its length and area is 18 square feet?
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Area=6cm2
perimeter=14cm

find the dimensions of the rectangle
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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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Anonymous commented
This is a question in my neice's math homework. It is in the chapter about Quadradic & Exponential Functions.
Draw a graph of a rectangle with length 1', Width 10', Perimeter 22', area 10'sq
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The length of a rectangle is 4 meters longer than it width.  If the perimeter of the rectangle is 60 meters, find the dimensions of the rectangle.
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The length of a rectangle is 6cm more than its width. If the area of the rectangle is 85cm^2, find the dimension
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The perimeter of a rectangular floor is 90 feet. Find the dimensions of the floor if the length is twice the width?
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A rectangle has sides that are 3x+1cm and 2x-5cm long. Its area is 39cm^2. What are the actual dimensions of the rectangle????
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The diagonal of frame is 5 inch. If the length of frame is 1 greater than it's width find the dimension of the frame?
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A rectangle has an area of 28cm2 and a perimeter of 22cm. What are the dimensions of the rectangle?

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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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Anonymous commented
Thank you
even though i wasnt the one to ask the question
this answer really helped me because it explains everything step by step
ill make sure to ask you my algebra questions in the future!
You can write a quadratic equation and solve it.
area = L*W
perimeter = 2(L+W)

area = L*(perimeter/2 - L) (substitute for W)
L2 - (perimeter/2)*L + area = 0 (rearrange to standard form)
L = (perimeter/2 ±√((perimeter/2)2 - 4(area)))/2 (use the quadratic formula)
L = perimeter/4 ± √(perimeter2/16 - area) (simplify)

Please note that there are two possible values for L. If you were to solve for W, you would get the same equation. This means that W is one of the values and L is the other. If you want L > W, then you will choose
L = perimeter/4 + √((perimeter/4)2 - area)
W = perimeter/4 - √((perimeter/4)2 - area)

Example
Suppose we know the perimeter of a 2x3 plot is 10, and its area is 6. Our formulas give
L = 10/4 + √((10/4)2-6) = 2.5 + √(6.25-6) = 2.5+.5 = 3
W = 2.5 - .5 = 2
_____
You will note that L = W = perimeter/4 = √area if the figure is a square.
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The length of a room, rectangular in shape, is 8 feet greater than its width. If the perimeter of the room is 64 feet, what are the dimensions of the room?
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