Same area, different perimeter

A paper chit 6 cm by 4 cm is cut into two equal pieces, each 6 cm by 2 cm. The whole chit has perimeter 2 × (6 + 4) = 20 cm. Join the two pieces end to end and you get a strip 12 cm by 2 cm, with perimeter 2 × (12 + 2) = 28 cm. The paper is the same, so the area is the same (24 sq cm), but the perimeter has changed.

The book asks for rectangles with whole-number sides and an area of 24 square units. There are four:

• 1 by 24: perimeter 2 × 25 = 50 units.
• 2 by 12: 28 units.
• 3 by 8: 22 units.
• 4 by 6: 20 units.

All have the same area. The long thin one, 1 by 24, has the greatest perimeter; the one closest to a square, 4 by 6, has the least.

Now take 9 unit squares (area 9 sq units), each joined to another along a full side. Put in a 3 by 3 square, the perimeter is 12 units, the smallest possible. Laid in a single row of 9, the perimeter is 2 × (9 + 1) = 20 units, the largest possible. The book's C shape (4 squares on top, 1 in the middle on the left, 4 at the bottom) also has perimeter 20 units.

What happens when you attach one more square? Each side of the new square either touches the figure (and hides a bit of edge) or faces outside (and adds a bit of edge).

• Touches along 1 side: the perimeter goes up by 2 units.
• Fits into a corner and touches along 2 sides: the perimeter stays the same.
• Fills a gap and touches along 3 sides: the perimeter goes down by 2 units.

So the area always grows, but the perimeter can increase, decrease or stay the same.