Sitting on the line
Faces are ten degree slices, so Gemini divides at ten and at twenty. The second of those divisions falls inside the spread of positions this date produces, which means the answer is year dependent in a way most birthdays are not.
Most years put the Sun past twenty degrees and into the final face. The years at the bottom of the range leave it in the middle face by a fraction. Neither outcome is unusual and neither is a problem; it is simply a case where a date is not enough on its own.
The same thing happens at sign boundaries, where it attracts far more attention. Here the stakes are smaller, since both outcomes leave the Sun in Gemini, but the arithmetic is identical and so is the remedy, which is to compute rather than to look up.
Why a date drifts at all
The tropical year is close to three hundred and sixty five and a quarter days, so each ordinary calendar year the Sun reaches a given date about six hours further round its orbit. The extra day in February then pulls the position back by roughly eighteen hours.
Averaged out, the drift is tiny. What remains is a spread of a degree and a bit across the whole of the last century, which is negligible almost everywhere and decisive here, because the boundary happens to sit inside it.
The two readings on offer
Below twenty degrees the Chaldean order gives the face to Mars and the triplicity count gives it to Libra and so to Venus. Above twenty the Chaldean order gives it to the Sun and the triplicity count gives it to Aquarius and so to Saturn, with Uranus added in modern practice.
Those are attributions from tradition rather than observed effects, and they should be read as such. What can be checked is the position itself, and once that is computed the choice of face follows from arithmetic rather than from opinion.
Both schemes date from long before anyone could compute a position quickly, which is part of why they were built as tables in the first place. A table is usable by hand, and a face is easy to find once a rough solar position is known from a calendar.
Beyond the face
The sign, meanwhile, is not in doubt at any point. Gemini runs to the June solstice, still more than a week away from this date, so there is no version of an 11 June birth that turns out to be Cancer.
The genuinely open questions are elsewhere. The Moon will be in one of two signs and the ascendant in any of the twelve, both decided by the hour. The widget above resolves the solar degree first, from your year, time and birthplace, and reports the margin to the nearest boundary.
Houses depend on the same missing information. Whether an 11 June Sun sits at the top of the chart, near the horizon or below the ground is decided by the hour of birth, and that placement is read as more consequential than the face question by most practitioners.
Questions people ask about this
Which decan is an 11 June birthday?
It varies by year. Most years place the noon Sun just past twenty degrees, in the third face of Gemini, while the lower end of the range leaves it in the second.
Is the sign ever in doubt?
No. Gemini holds for this date in every year sampled at noon from 1930 to 2035, with the crossing into Cancer more than a week later.
What is the solar degree on 11 June?
Roughly twenty degrees, with a noon range of about nineteen point six to twenty one across the years checked.
Does the birth hour matter here?
It can. The Sun moves about a degree a day, so in a year that lands close to twenty degrees the time of day decides which face applies.
What are the rival decan rulers?
Mars or Venus for the second face and the Sun or Saturn for the third, depending on whether the source uses the Chaldean order or the triplicity count.
Should I read both faces?
Some practitioners do when a position is this close to a line. Others compute the degree, take the face it falls in and leave it there. Both are defensible, since neither table is a measurement.