I got the multiplication precision rule wrong:

relative precision is defined as the uncertainty divided by the magnitude of the measured quantity, e.g.

10 meters +/- 1 meter gives an uncertainty of 1 meter, and a relative precision of 1/10 or 10%

The rule is that the relative precision of the multiplication is the sum of the relative precisions of the factors.

When we square 10 +/- 1 meters to find the square area corresponding to that side, we get 100 square meters with a relative precision of

1/10 + 1/10

or a 20% relative precision of the 100 square meter result, which squares (sorry) with the +/- 20 range computed by squaring the range 9 through 11 meters for a squared area range of 81 through 121 square meters.

The problem still remains, though, that the midpoint of the squared range is 101, and not 100!