symmetry vs balance

symmetry and balance both are nouns.

Word	Noun	Adjective	Verb	Adverb
symmetry	Yes	No	No	No
balance	Yes	No	Yes	No

As nouns, symmetry and balance are synonyms defined as:

symmetry and balance: (mathematics) an attribute of a shape or relation; exact reflection of form on opposite sides of a dividing line or plane

Other synonyms of symmetry include correspondence, symmetricalness.

As nouns, balance is a hypernym of symmetry; that is, balance is a word with a broader meaning than symmetry:

Other hypernyms of symmetry include counterbalance, equilibrium, equipoise.

symmetry (noun)	balance (noun)
(mathematics) an attribute of a shape or relation; exact reflection of form on opposite sides of a dividing line or plane	a scale for weighing; depends on pull of gravity
(physics) the property of being isotropic; having the same value when measured in different directions	a weight that balances another weight
balance among the parts of something	a wheel that regulates the rate of movement in a machine; especially a wheel oscillating against the hairspring of a timepiece to regulate its beat
	harmonious arrangement or relation of parts or elements within a whole (as in a design)
	(mathematics) an attribute of a shape or relation; exact reflection of form on opposite sides of a dividing line or plane
	equality between the totals of the credit and debit sides of an account
	the difference between the totals of the credit and debit sides of an account
	something left after other parts have been taken away
	equality of distribution
	a state of equilibrium

symmetry (verb)	balance (verb)
	hold or carry in equilibrium
	compute credits and debits of an account
	bring into balance or equilibrium
	be in equilibrium