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Important to note are these formulae:

Perimeter_of_rectangle = 2 x (length + width)

Area_of_rectangle = length x width

So if the perimeter and area are known, then:

2 x (length + width) = perimeter

=> length + width = perimeter / 2

=> length = perimeter / 2 - width

length x width = area

=> (perimeter / 2 - width) x width = area (substituting for length given above)

=> perimeter / 2 x width - width2 = area

=> width2 - perimeter / 2 x width + area = 0

which is a quadratic and can be solved either by factorization or by using the formula:

width = (perimeter / 2 +/- sqrt(perimeter2 / 4 - 4 x area)) / 2

= (perimeter +/- sqrt(perimeter2 - 16 x area)) / 4

This will provide two values for the width. However, each of these values is the length for the other, so the larger value is the length and the smaller value is the width.

Sometimes only 1 value will be found for the width above. In this case, the rectangle is actually a square which means that the length and width are both the same.

Examples:

1. perimeter = 6, area = 2

width2 - perimeter / 2 x width + area = 0

=> width2 - 6 / 2 x width + 2 = 0

=> width2 - 3 x width + 2 = 0

=> (width - 2) x (width - 1) = 0

=> width = 2 or 1.

So the length is 2 and the width is 1.

2. perimeter = 12, area = 9

width2 - perimeter / 2 x width + area = 0

=> width2 - 12 / 2 x width + 9 = 0

=> width2 - 6 x width + 9 = 0

=> (width - 3)2 = 0

=> width = 3

So the rectangle is a square with both length and width of 3.

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Q: How do you calculate the top dimensions of a rectangle if you know its perimeter and bottom area?
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