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How did Archimedes find the area of a segment of a parabola?

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Portrait of Archimedes by Giuseppe Nogari shows an elderly man with white hair and beard, holding a compass, gazing downward, set against a dark background. — Archimedes Archimedes, oil on canvas by Giuseppe Nogari, 18th century; in the Pushkin Fine Arts Museum, Moscow. Archimedes was an ancient Greek mathematician and inventor.

Archimedes determined the area of a parabolic segment using two distinct approaches: a heuristic "mechanical" method and a rigorous geometric proof.12

The Mechanical Method

In his work Method Concerning Mechanical Theorems, Archimedes described a process of discovery where he divided the parabolic segment and a corresponding triangle into an infinite number of infinitesimally thin strips. By "weighing" these strips against each other on a notional balance, he determined the ratio of the two figures. While this technique was essential for his discovery, he acknowledged that it did not constitute a formal mathematical proof.2

The Geometric Method

In his treatise Quadrature of the Parabola, Archimedes provided a rigorous proof using the method of exhaustion, a technique originally developed by Eudoxus of Cnidus. He demonstrated that the area of any parabolic segment is exactly $4 / 3$ the area of a triangle with the same base and height. He achieved this by calculating the sum of an infinite geometric series:12
$1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \dots = \frac{4}{3}$
By successively adding triangles to fill the segment, he showed that the area could be "exhausted," with $4 / 3$ serving as the limit of the partial sums.1