20 Feb '15 18:222 edits

Recall that "unit circle" means a circle with a

Two unit circles can be positioned so that they touch in tangent fashion at just a single point. You can place a third unit circle so that it touches the first two circles in tangent fashion.

Doing this creates a "pocket" region in between the three unit circles.

Q1: What is the area of the pocket?

Next we can find a small circle (radius < 1) that will fit inside the pocket so as to touch the three unit circles in tangent fashion. This creates three small regions that are between the four circles, not interior to any of them.

Q2: What is the combined area of those three small regions?

My source derives the answers by using good old schoolboy geometry-- no calculus needed. It gives not just a decimal approximation (such as you could work up by drawing this in CADD) but the precise answers. (Hint: irrational numbers come into play.)

Q3: My source stops there, but you could continue this process by adding three circles smaller yet, and then three more even smaller ones, and so on. In the limit of adding an infinite number of ever-smaller circles, what is the area trapped between all the circles?

{I suspect Q3 is very time consuming to answer, and I may not be competent to decide if an answer you give to Q3 is correct. I just throw Q3 out there in case anybody wants to

*radius*of one.Two unit circles can be positioned so that they touch in tangent fashion at just a single point. You can place a third unit circle so that it touches the first two circles in tangent fashion.

Doing this creates a "pocket" region in between the three unit circles.

Q1: What is the area of the pocket?

Next we can find a small circle (radius < 1) that will fit inside the pocket so as to touch the three unit circles in tangent fashion. This creates three small regions that are between the four circles, not interior to any of them.

Q2: What is the combined area of those three small regions?

My source derives the answers by using good old schoolboy geometry-- no calculus needed. It gives not just a decimal approximation (such as you could work up by drawing this in CADD) but the precise answers. (Hint: irrational numbers come into play.)

Q3: My source stops there, but you could continue this process by adding three circles smaller yet, and then three more even smaller ones, and so on. In the limit of adding an infinite number of ever-smaller circles, what is the area trapped between all the circles?

{I suspect Q3 is very time consuming to answer, and I may not be competent to decide if an answer you give to Q3 is correct. I just throw Q3 out there in case anybody wants to

*really*strain the brain. Don't beat your head against the wall on this one if you think it is equivalent to writing a master's thesis or something. 😛}