Mathematical Marvels of Pythagoras

Welcome, fellow puzzle enthusiasts! Pythagoras' Triangle Logic Puzzles are the ultimate test of your mathematical mettle. Can you solve these brain-twisting conundrums?

Triangle Formation

A triangle has three sides, a, b, and c. Given that a + b + c = 12, and a^2 + b^2 = c^2, what is the area of the triangle?

Hint: Think about Pythagorean triplets!

Solve it!