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How does Heron's formula calculate the area of a triangle from its side lengths?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

Heron’s formula is a method used in geometry to calculate the area of a triangle when the lengths of all three sides—a, b, and c—are known.12

To apply the formula, you first determine the semi-perimeter (s), which is half of the triangle's perimeter:
$s = \frac{a + b + c}{2}$

Once the semi-perimeter is found, the area (A) is calculated using the following equation:
$A r e a = \sqrt{s (s - a) (s - b) (s - c)}$

This formula, credited to the Greek mathematician Heron of Alexandria, provides an alternative to the standard method of multiplying one-half of the base by the altitude.13