# Are similar circles really a thing?

I'm a fifteen year old who is currently studying circle geometry (if that is the appropriate term) and our teacher stated that concentric circles are similar. I thought about this, and it doesn't make sense to me. The reason is because of proportionality. For example, similar triangles are similar because they have the same angles and they have proportional sides. However, circles can not be compared for angles, so that's out (as they all have the same 360 degree angle at the center) and the only factor is their size, which is directly influenced by their radius. If the radius is the only variable involved in a triangle like this, how can a circle be NOT proportional to another circle? If a case of that existed, there would be meaning (at least from my current perspective) to the term "similar circle."

Help and critique on my logic is requested, and an explanation as to the term "similar circle."

• Concentric circles are similar. So are non-concentric circles. – André Nicolas Mar 4 '16 at 5:06
• But what is the point of the term "similar circle" if there are no cases of "non similar circles"? I'm really confused about this topic. – Gil Keidar Mar 4 '16 at 5:07
• In a proof, one might be using the fact that (any) two circles are similar, so it may be useful to mention it. – André Nicolas Mar 4 '16 at 5:09
• You are right, in Euclidean geometry all circles are similar. Likewise, all parabolas are similar. – bof Mar 4 '16 at 5:10
• "similar" does not only refer to circles. It refers to geometric shapes. For example, it doesn't make so much sense to speak of a rectangular square although a square is always rectangular. – Friedrich Philipp Mar 4 '16 at 5:12

## 2 Answers

You're right: any two circles are similar (and so there's not much point of talking about "similar circles")! In general, two shapes are "congruent" if you can turn one into the other by translations (moving around in the plane), rotations, and reflections. Two shapes are "similar" if you can rescale ("zoom in or out" on the picture) one of them to turn it into a shape that is congruent to the other. Given two circles, you can rescale one so that it has the same radius as the other, and then any two circles with the same radius are congruent since you can just translate the center of one to the center of the other.

• Nice explanation, appreciated by the OP. Just to whet his eighth grade appetite: if you think about geometry on the surface of a sphere you'll see that a pair of similar triangles must in fact be congruent. There's no "zooming in or out". – Ethan Bolker Mar 4 '16 at 13:42
• @EthanBolker Don't confuse him - nonEuclidean geometry is neither useful to him nor relevant to him now... – Daniel Mar 4 '16 at 17:10
• @Daniel The OP has asked some other interesting questions and is mathematically adventurous for his age. I'm not sure when (spherical) nonEuclidean geometry is either particularly useful or relevant (other than for navigation?) but it can be eye opening. If he finds my suggestion intriguing he can follow it and probably won't be confused by it. Or he can ignore it. – Ethan Bolker Mar 4 '16 at 18:07
• Oh I have nothing against mathematical exploration. I'm 17 and I love learning new things, but I know that when I was 15 and taking geometry, spherical geometry just about drove me insane. I don't think I was ready for it at the time. He very well may be ready, but you should at least provide a disclaimer... – Daniel Mar 4 '16 at 18:18
• I just happened to see a really good video on this yesterday by Matt Parker, "There is only One True Parabola": youtube.com/watch?v=hoh4TmPzu1w – rrauenza Mar 4 '16 at 23:24

Yes indeed. Every circle is similar. You can always scale one of them to match the other. Actually, this is the definition of similarity. In case of triangles, this definition yields the result that the sides are proportional. "The sides of one triangle are proportional to the other" is not the actual definition of similarity. You may have a look here