Same Area, Different Perimeter

One boy's personal exploration into Infinity

We'd been doing all that investigation into 2D shapes, looking at area and perimeter. We had  found shapes that had the same perimeter but different area. You may have read about it on a previous post.

This provoked one boy to inquire in the opposite direction.

"What if I keep the area the same, but change the length of the sides?" he mused.

Not content with that, he continued, "...and what if I use triangles instead of rectangles?"

Here is the page of diagrams that he drew to explore this idea:

He started with a right angled triangle with sides 8cm and 4cm = an area of 32 square cm (Sorry - I don't think I can do superscript for index notation on this text editor.)

Then he doubled the long side and halved the short side: 16cm x 2cm = 32 square cm

Realising that he would go off the page the next time he doubled the dimensions of the long side, he decided to use a scale of 1:2 for the next diagram:  32cm x 1cm = 32 square cm

His Stunning Conclusion

"You know what?" he asked. "I think there is an infinite number of triangles with an area of 32 square cms. I could keep on doubling and halving forever."