Geometry Lab reference
Heron’s formula
Find a triangle’s area using only its three sides.
Area = √[s(s − a)(s − b)(s − c)]
The idea
The letter s is the semiperimeter, or half the distance around the triangle: s = (a + b + c) / 2. Heron’s formula works for any valid triangle when all three sides are known.
Work through an example
For sides 3, 4, and 5, s = 6. The area is √[6 × 3 × 2 × 1] = √36 = 6 square units. The height to side c is then 2 × area / c = 12 / 5 = 2.4 units.
What to watch for
The side lengths must first satisfy the triangle inequality: any two sides add up to more than the third. A flat triangle has zero area and is excluded by our solver.
Make the formula move.
Explore this idea with real inputs.