A date is a place in the orbit
The zodiac used here is tropical, which means it is measured from the equinox rather than from the stars. Zero degrees of Aries is the March equinox point, and every other longitude is counted forward from there. The Sun's longitude is therefore just another way of saying how far through the seasonal year a moment falls.
Calendar dates are trying to say the same thing. Where the two agree, the Sun's position on a given date repeats. Where they disagree, it slips. The Gregorian calendar was designed to keep the disagreement small, and it does.
What the leap rule actually does
A tropical year is close to three hundred and sixty five and a quarter days, so a common year of three hundred and sixty five ends about six hours early against the orbit. Three of those in a row leaves the calendar most of a day ahead, and the leap day in February takes it back. The pattern is a sawtooth rather than a slide.
On 29 August the visible result is a noon solar longitude that has stayed between about 5.2 and 6.6 degrees of Virgo since 1930 and will stay there through 2035. The century rules that drop the leap day in years like 1900 and 2100 nudge the pattern over longer spans, but not within a lifetime.
The rest is a matter of arithmetic on your own details. The year sets the base position, the hour adds a few arcminutes for each hour past midday, and the birthplace decides how the local clock reading maps onto universal time.
Six degrees, and what sits with it
Six degrees of Virgo falls in the first of the sign's three ten degree faces. In the Chaldean rulership order that face belongs to the Sun, while the triplicity order gives it to Virgo and therefore again to Mercury, which rules the sign itself. Descriptions of this decan differ sharply depending on which scheme an author started from.
The degree itself carries no traditional meaning independent of the face and the aspects made to it. What it is good for is precision. A planet at six degrees of Taurus or Capricorn trines this Sun closely, one near six degrees of Pisces opposes it, and none of those relationships can be read off a sign name alone.
Steady Sun, unsteady everything else
The stability applies to the Sun and stops there. The Moon on this date differs entirely from year to year, since its cycle of about twenty seven and a third days will not divide into a calendar year, and within a single 29 August it moves about thirteen degrees, so it can change sign partway through the day.
The Ascendant repeats on a daily cycle rather than a yearly one, so it depends on the hour and the place and not at all on the date. Two people born a century apart on 29 August share a solar degree and no more than that. The widget above will show you where your own Sun landed inside the band and how far it stood from the sign's edges.
It is a short calculation and the answer does not change once it is done.
Questions people ask about this
Is 29 August always a Virgo birthday?
Yes, in every year from 1930 to 2035 at noon, and by a wide margin. The Sun entered Virgo about a week earlier and has another three weeks to run before it leaves.
Does my birth year change my sun sign?
Not here. The year moves the solar longitude by a fraction of a degree either way, which is nowhere near enough to cross a boundary thirty degrees wide from the middle of a sign.
Why does the Sun return to the same degree each year?
Because the calendar is built around the orbit. The quarter day the year overruns is absorbed by a leap day every four years, so a fixed date keeps returning to nearly the same point rather than sliding through the seasons.
What decan does 29 August fall in?
The opening ten degrees of Virgo, with the Sun around six degrees. That face answers to the Sun in the Chaldean order and to Mercury in the triplicity order.
Can the position on this date ever drift further?
Over many centuries, yes, because the leap rule is an approximation and the skipped leap years at most century marks shift the pattern. Across the span of family records it is not a practical concern.
How precise is the figure the widget gives?
It works from your year, your local birth time and the coordinates of your birthplace, and returns the longitude to minutes of arc along with the distance to the nearest sign boundary.