# A Circle Toy

Here’s a fun little toy.

It’s useful for building an argument about the area of a circle.

Let’s say you know that π (pi) is the ratio of the circumference of a circle to its diameter. Then, because the diameter is twice the radius, you know the circumference is 2π r.

Usually this argument proceeds by drawing pictures. It turns out to be really satisfying to get your hands on this piece of mathematics.

Pull the circle apart.

Each semi-circle has an arc that measures π r. Straighten them out and reassemble.

Now the same area has been reassembled into something that is approximately a parallelogram. If you treat it as a parallelogram, its base would be π r, and its height would be r, so its area would be π r².

Please note that this argument—whether in hands-on toy form, or diagram form—is not a proof that π r² is the area of a circle with radius r. It does suggest that the area is approximately that. But you’ll need to imagine smaller (like really really small) sectors, and you’ll need a little calculus to make it all rigorous.

But it’s a nice argument, and it’s a super fun toy.

You can make your own using this SVG file (which contains one full circle).

### Extension

Let’s say you double the radius of your circle. How does the area grow?

Well, now you’ve got a quarter of the larger circle (approximately, just like that parallelogram earlier wasn’t really a parallelogram), so you’ve grown the area by a factor of four.

What if you triple the radius?

Here things get a little funny. A third half-circle spans a quarter of the way around.

Then you’ll need another half circle with smaller sectors (these are 32nds of a circle instead of 16ths in the original). And you’ll still need some more small sectors. How many more? Eight, or half of a half-circle’s worth.

Total: 4.5 half-circles, so 18 to make the whole thing. If you triple the radius, the area grows by a factor of nine.

We were inspired to make this model following a conversation with Amy Shell-Gellasch. An old-school version of this model, together with a host of other wonderful old math learning aids, resides in the Smithsonian Institution. You can browse those objects online!

### Notes on making the model

Our file is made for cutting the sectors in the parallelogram arrangement. That is intentional. This way, each vertex is at the intersection of three curves—the two straight line segments, and the curvy boundary. If you cut in the circular arrangement, the laser visits that vertex 8 times and you end up with a lot more charring.

We use ½-inch thick basswood or balsa. Doing this requires a special technique we call the Varat Flip, after our laser specialist Jordan Varat. Jordan and Executive Director Lauren Siegel developed this method for making one pass each from the front and back of a thicker piece of wood.

Finally, we have used ½-inch wide gaffer tape to hold everything together. But we have also found the stickiness of gaffer tape to vary greatly and we don’t have a reliable source for extra-sticky gaffer tape. We recently discovered that ½-inch wide duct tape exists, and we’ve been very happy with the results.