What is the name for a shape with ten sides?

A decagon, from the Greek deka. The naming runs on Greek numerals throughout - pentagon, hexagon, heptagon, octagon, nonagon, decagon - with the nonagon breaking the pattern by being half Latin, which is why purists prefer enneagon and nobody else does.

Above about twelve sides mathematicians generally stop naming and start counting, writing 17-gon rather than heptadecagon. The exception is the ones with a story: the 17-gon has one, because a regular version can be constructed with compass and straightedge alone - which nobody had suspected for two thousand years until Carl Friedrich Gauss proved it in 1796, aged nineteen. He was so pleased that he asked for one on his gravestone. The stonemason declined, on the grounds that at that many sides it would be indistinguishable from a circle.

That result also settled which regular polygons are constructible at all, and the answer is a strange and specific list tied to a rare kind of prime number. A 7-gon or a 9-gon cannot be drawn with compass and straightedge no matter how clever you are - which is the same class of impossibility as trisecting an angle or squaring the circle, and it took until the nineteenth century to prove rather than merely fail at.