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To find the number of rectangles with a given perimeter ( P ), you can use the formula for the perimeter of a rectangle, which is ( P = 2(l + w) ), where ( l ) is the length and ( w ) is the width. Rearranging gives ( l + w = \frac{P}{2} ). If ( P ) is an even number, the number of integer pairs ( (l, w) ) that satisfy this equation is determined by how many ways you can express ( \frac{P}{2} ) as a sum of two positive integers. Each pair represents a unique rectangle, so you can find the count by determining the integer partitions of ( \frac{P}{2} ) minus one for each dimension being equal.

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1w ago

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