I first saw this video over at MathFail. It’s cute, and I have to admit, kind of cool.

But anything this simplistic always sends me into skeptical mode.

Before we go any further, check it out:

Do you believe it?

This might be the question to start a discussion with a student. It’s certainly the first question that comes to my mind.

If it were really this simple, wouldn’t we have used it to “prove” the formula for the area of a circle much earlier?

What’s wrong with it?

For the “proof” in the video to work, you have to assume (or believe) that the circumference is 2πr. This seems a bit cheesy to me, since that formula is as complex as the one we’re trying to prove. Not to mention quite closely related. But I’ll let this one go.

The thing that really bothers me is that they use only a few chains – each of which has thickness.

If you filled the inside of a circle (a disk) with concentric circles, none of those circles would have a thickness. In fact there’s an infinite number of those circles.

Is it realistic to take each of those circles and fold them out and get a triangle?

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4 Responses to Area of a Circle vs. Area of a Triangle

I liked this video a lot when I first saw it. Yes, it’s not a rigorous proof that I had to learn in school. But I think it’s a great starting point for a conversation. And it’s a lot less scary. So, as an intro or a supplement to the topic it works. Plus I can see how it can open up a conversation about how close should a “close enough” be, precision, and approximation error.

I liked this video a lot when I first saw it. Yes, it’s not a rigorous proof that I had to learn in school. But I think it’s a great starting point for a conversation. And it’s a lot less scary. So, as an intro or a supplement to the topic it works. Plus I can see how it can open up a conversation about how close should a “close enough” be, precision, and approximation error.

Good point, Yelena. Conversation starters are always welcome!

If you formulate it like Archimedes did (Proposition One on the page below) you do not need to assume anything about the circumference.

http://en.wikipedia.org/wiki/Measurement_of_a_Circle

Thanks so much for that link. I had no idea!