As you know, this is a special type of rectangle called a square.

If you walked the perimeter of this rectangle you would travel 4 + 4 + 4 + 4 = 16 units of length.

The area of this rectangle is 4 x 4 = 16 square units.

The units for perimeter and area are different but both measures equal 16.

For what other rectangles are the measures for perimeter and area equal?

Using squared paper, the student can investigate if other rectangles are possible. They are likely to try many unsuccessful rectangles before they find one that works.

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A more systematic approach is to constrain the length of one side and experiment with the other. Below the length is set at 10 units and various rectangles tried until the conditions are met.

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Students may begin by defining the unknown sides as l (length) and w (width) and finding equations to work out perimeter and area.

Perimeter = l + w + l + w = 2 (l + w)
Area = l x w or lw

Since perimeter and area must be equal the single equation becomes:

2(l + w) = lw

Having established the equality students can set l and solve for w or vice versa to find solutions.

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In only three solutions do both l and w have whole number values. Students can generate a set of solutions and organise the data in a table.

Students might also recognise that the denominator of w is always l – 2. The numerator of w is always double l.

This yields the general formula of w=2l/(l-2). The formula can be found be rearranging the equality formula though the although the algebra is about Level 6 or 7 of NZC.