A voxel world has no curves available. Building a circle means choosing which blocks in a square grid belong to it, and the standard rule is the obvious one: include a block when its centre lies within the radius.
include (x, y) when (x − centre)² + (y − centre)² ≤ r²
That is the whole algorithm, and it is why every circle generator agrees with every other one. The Minecraft circle generator applies it and reports the grid and the block count.
The count is not the area
A continuous circle of diameter 9 has an area of 63.6. The block version takes 69. That is 8 per cent more material than the geometry says, because every block that is more than half inside gets counted whole and the rounding does not cancel out at small sizes.
It does cancel eventually. At diameter 15 the count is 177 against an area of 176.7, which is close to exact. At 21 it is 349 against 346.4. By 31 the count is 749 against an area of 754.8, so the error has crossed over and the block circle now uses slightly less than the area predicts.
This is the usual behaviour of a discretised boundary. The error lives on the edge, the edge grows with the diameter while the area grows with its square, and the relative error therefore shrinks towards nothing. For planning purposes, use the generator below about 20 blocks and pi r squared above it.
Odd diameters build better
An odd diameter puts a single block at the centre, so the circle has a true middle and four clean axes of symmetry running through block centres. An even diameter centres on a corner between four blocks, which produces a shape that is symmetric but has no centre block and tends to look slightly wrong where the axes meet.
It matters most for anything you intend to build outwards from, such as a dome or a tower, where the centre column needs somewhere to stand. Most builders default to odd diameters for this reason without necessarily knowing why.
Arches are the same idea in one dimension
An arch is a circle problem with the vertical collapsed. For each column across the span you want the height of the curve above it, which is
height = round(sqrt(r² − (x − midpoint)²))
Nine blocks wide gives a centre height of 5 and takes 31 blocks, which the arch generator works out column by column. The rounding is doing the same job as before: turning a continuous curve into a staircase, with the steps bunched at the ends where the curve is steepest.
And then you have to carry it
Having worked out that you need 749 blocks, the next question is how many trips that is. Most items stack to 64, which is a power of two and no accident, and inventory arithmetic follows from it. A single chest holds 27 slots, so 1,728 items. A double chest holds 54 slots and 3,456.
So a 31-block circle is 11 stacks and 45 blocks, a little over two fifths of a single chest. The stack calculator converts in both directions, which is more useful than it sounds when the answer determines whether you need one trip or four.
For the other quadratic the game runs on, see the piece on levelling curves, where Minecraft's experience system stitches three of them together.