I have a confession to make. One that will lower me seriously in the esteem of my 10-year-old nephew: I don't really get Minecraft.
Sorry, buddy. I've tried it. I love that you love it -- honestly, creative games are awesome for your problem-solving skills and breaking down the barriers between art and science, so I'm all for it in principle -- it's just not my cup of tea.
All the same, I'm happy to help you solve problems! Of course I am. That's what uncles are for.
So, you want to make a circle? Awesome. One tiny problem: the Minecraft world is made of squares, and you can't make a perfect circle out of squares. You can, however, make something that looks pretty much like a circle -- that's what happens when your computer draws a circle; using lots of tiny dots means you can get close enough!
So, how would you draw a circle that was, say, 21 squares wide? That would be a circle with a radius of 10 units.
It's easy to find a few places blocks need to go: ten squares north of the centre of the circle, you'll have a block. Same thing 10 squares east, south and west of the centre -- four points making a sort of cross with the centre.
After that, it gets trickier -- how about a square to the north-east of the centre? It needs to be 10 units away from the centre, so you'd think you'd go ten squares diagonally -- but unfortunately, that's too far! That would make a big square. Why is it too far? It's because a diagonal step is bigger than a north/south/east/west step. It's not twice as far, though: if you go five squares to the northeast of the centre, you end up midway between your compass points, which isn't right, either. The right answer is somewhere in between five and ten.
It turns out to be seven squares1, because of Pythagoras's Theorem. You might have come across this, you might not: in a Minecraft context, it says, if you go east one number of squares and north another number, you can work out how far from the centre of the circle you are by using the recipe:
How does it help? Well, if you go seven squares across and seven squares up, following those steps gives you 49 and 49, which add up to 98. The calculator says the square root of 98 is about 9.9, which rounds up to 10 -- the square seven east and seven north of the centre is pretty close to 10 units away from it, which is what we want from a circle with radius 10!
We can change the recipe slightly to work out how far north we need to go if we go any distance to the east. Here's how it looks:
If you go one to the east, the steps give you 1, 100, a difference of 99, and the square root is 9.95 or so -- which rounds up to 10, so the square one to the east and 10 north is on the circle. You can write this efficiently as (1, 10).
Working the same way for the next few numbers gives (2, 10), (3, 10), (4, 9), (5, 9), (6, 8) and (7,7) -- and surprisingly, that's all the hard work we need to do! We can now use the idea of symmetry to find all of the other coordinates.
One kind of symmetry means you can swap the numbers in a pair of coordinates: because (1, 10) is on the circle, so is (10, 1). In the same way, you know (10, 2), (10, 3), (9, 4), (9, 5) and (8, 6) are also on the circle.
Now you've got a quarter-circle! (I believe, in Minecraft, that's enough; you can set it up to build the whole circle using symmetry. But let's keep going!)
How do you get the rest of the circle? Simple. The coordinates count how far east and how far north you need to go -- but you can switch the directions around! Instead of going east and north, you could go west and north, or east and south, or west and south -- making up the four quarters of the circle! (The important thing to remember is that you always need one sideways direction and one up/down direction for it to work.)
I hope that's enough for you to get going -- and that it's practical enough for you to draw your circle!