Here is a better version of his figure:

The point S is the center of the large semicircle, as P and Q are the centers of the smaller semicircles; so ST, as I’ve labeled it, is the radius, a+b, while AT is the radius r_{1}. Therefore AS = ST – AT = a + b – r_{1}.

Let me know if there are any other parts to be clarified.

]]>I have a question. Why AS^2=(a+b-r1)^2 ? ]]>

I agree that teachers probably cover “diamonds” because they are a different shape that would be of interest to children, and they want to use terms that are familiar to them.

The problem is that the word is ambiguous; in a sense it is not “a unique shape” (not the sense you intended!), but has several different meanings, so that it can become confusing.

As Doctor Ian pointed out, one of those senses is “a square rotated 45 degrees”; but he said it in the way he did (“‘becomes’ a diamond”) to emphasize that this is not a proper way to talk about shapes. Doctor Rick pointed out that the concept of shape should be independent of orientation; that is, rotating a figure can’t really change its shape. The rotated figure is still a square! So that “definition” of “diamond” is not a proper mathematical definition; it is just one of the “everyday” uses that children are exposed to. Avoiding such confusion is one of the main reasons we are suggesting that the term “diamond” should not be used as if it were a technical term. Just teach kids the name “rhombus”, and you get to discuss this interesting shape without raising unnecessary questions.

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