As a child, I remember being mystified how a circle could hold any great secrets. After all circles are so easy to construct and they look the same from all directions.

How wrong I was!

In studying circles, as with investigating anything of importance, one question leads to another, and insights gained in other areas of mathematics prove to be relevant to our study and reveal yet more hidden patterns. Entire books have been written about the geometry of circles, and even more books about π, the ratio between a circle’s circumference (perimeter) and diameter.

I remember being astounded when I learned that the size of the angle at the centre of a circle (the yellow angle in this image) is always exactly twice the size of ANY blue angle on the circumference! The ratio was always two, and the conclusion was that it did not matter where on the circumference you placed the angle, it would *always* be the same size … if it was standing on the same arc. Such amazing simplicity! And so, my adventure with circles began.

As you will see, there is so very much more to discover …

Brilliant!!! This helps so much!!! So much easier than anything else I have found. This is now simple.

Thanks again for showing the simple way of doing this. I look forward to the next video!! Absolutely brilliant.

Once again, excellent! I’ve watched many others, that have not helped nearly as much as these. Thanks for making it so easy to understand. When things get complicated, it is easy to make a mistake, but your method of writing down the structure helps to prevent mistakes.

Thanks a million!!! I’ve watched many videos and read tutorials, but still could not get two in a row correct on Khan Academy. Your explanation was so clear and simple that it finally made sense to me and now I can get all of the problems correct!!! Thank you very much.

It all makes total sense now! Thanks very much for all the derivative videos!!! Very helpful.

MemorizeAndLearn (on four different CCM YouTube videos about Differentiation)

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