ax vs hatchet

ax and hatchet both are nouns.

Word	Noun	Adjective	Verb	Adverb
ax	Yes	No	Yes	No
hatchet	Yes	No	No	No

As nouns, hatchet is a hyponym of ax; that is, hatchet is a word with a more specific, narrower meaning than ax:

ax: an edge tool with a heavy bladed head mounted across a handle
hatchet: a small ax with a short handle used with one hand (usually to chop wood)

ax (noun)	hatchet (noun)
an edge tool with a heavy bladed head mounted across a handle	a small ax with a short handle used with one hand (usually to chop wood)
	weapon consisting of a fighting ax; used by North American Indians

ax (verb)	hatchet (verb)
terminate
chop or split with an ax