80 cm and 15 cm gives you the tangent of the angle involving the stick and the hypoteneuse. That angle is the same as the one that tells you how far off of verticle the sun is at noon. If you know the altitude of the sun at noon on the solstice (or any other day where you know the Sun's declination) you can work backward to your latitude.

You can use the angles for both solstices to split the difference and get the angle for an equinox, though. The equinox angle is your latitude angle. This won't tell you if you are north or south of the equator though. 8)

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-al