Caption: "Geometry of a total solar eclipse." (Slightly edited.)
Features:
However, its umbra is much smaller than a quarter of the Earth's diameter.
The image shows why this is so diagrammatically.
Analysis of umbra size is given in the IAL subsection Umbra Size.
But we do not need that analysis here. We just take the small Moon's umbra as a given.
Solar eclipses can happen when the Moon is at any of its distances during an almost exact nodal algnment (during an eclipse season): but total solar eclipses only when the Moon is relatively near and its umbra touches down on the Earth.
Note the Sun has no sharp edge. The solar atmosphere just morphs into the solar wind as one moves outward.
However, there is a relatively sharp layer from which most light escapes the Sun. That is the solar photosphere.
One should NEVER look at the solar photosphere even the smallest part thereof without proper protection: i.e., a valid astronomical solar filter. We're always catching little glimpses of the solar photosphere, and so one shouldn't be paranoid about this. But minimizing those glimpses is best.
Only during totality is it safe to look at the Sun with the naked eye.
In this case, if you are located below the tip of the umbra's cone, you see the night side of the Moon centered on the Sun with a bright ring (i.e., an annulus) of the solar photosphere surrounding it.
Since part of the solar photosphere is visible, you should NOT look at the Sun without proper protection: i.e., a valid astronomical solar filter.
Both of these distances vary due to the eccentricities of the Moon's orbit (mean value e = 0.0549006 ∼ 5.49 %) and the Earth's orbit (e = 0.0167086 ∼ 1.67 %: J2000).
So as NOT to go into complexities, we'll just say the Earth-Moon distance has to be a little less than its average value (384,748 km = 60.3229 Earth equatorial radii (R_eq_⊕ = 6378.1370 km)) for total solar eclipses since total solar eclipses are a little less frequent than annular solar eclipses.
The frequency of solar eclipse types during eclipse seasons for calendar years 2000 BCE--3000 CE is total solar eclipses 26.7 %, annular solar eclipses 33.2 %, hybrid solar eclipses 4.8 %, partial solar eclipses 35.3 %, and total solar eclipses plus hybrid solar eclipses 31.5 % (see Fred Espenak: MrEclipse.com: scroll down ∼ 60 %). Hybrid solar eclipses flick between being marginally total solar eclipses and annular solar eclipses.
But if the surface is NOT perpendicular to the umbra axis, the umbra tip could be stretched out as shadows are at sunset.
Presumably, this stretching out effect is accounted for in the 267 km at most value reported above. But I cannot find a discussion of this fine point.
v = 1.022 - 0.46511 = 0.5569 km/s = 33.41 km/m = 2005 km/h .
The actual minimum eclipse path velocity varies a bit depending on various factors and is more typically ∼ 1700 km/h (see Wikipedia: Solar eclipse: Occurrence and cycles).
The maximum fiducial eclipse path velocity occurs at the Earth's poles where the Earth's tangential velocity is zero. This maximum value is, of course 1.022 km/s = 3679 km/h.
To first order, the diameter of the region is just that of the Moon: 3464.2 km ≅ 0.273 Earth mean diameters (see Wikipedia: Moon).
This size is obtained by just assuming the Sun's rays are all parallel, the Earth is a flat disk, and all of the Moon's penumbra lands on the Earth.