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[Edit] OTHERS HAVE GIVEN THIS ANSWER, BUT I CAN'T DELETE IT.
There are trig functions that are known to equal π. Their series expansions are used to approximate π to the accuracy desired, but they also prove that the value of π is 3.14159...
For instance, knowing that π/2 = arcsin( 1 ) means that you can use the Taylor series of arcsin to prove the value of π to any accuracy you want. I don't know if this example is practical, but there are many other similar ways. See https://en.wikipedia.org/wiki/Pi#Infinite_series
There are trig functions that are known to equal π. Their series expansions are used to approximate π to the accuracy desired, but they also prove that the value of π is 3.14159...
For instance, knowing that π/2 = arcsin( 1 ) means that you can use the Taylor series of arcsin to prove the value of π to any accuracy you want. I don't know if this example is practical, but there are many other similar ways. See https://en.wikipedia.org/wiki/Pi#Infinite_series