Heron's formula
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Sometimes you can measure all three sides of a triangle — a triangular field, a sail, a roof truss — but you have no easy way to measure a height. Heron's formula, named after Heron of Alexandria, computes the area from the three side lengths alone, with no height and no angle needed.
First find the semi-perimeter s = (a + b + c) / 2, that is half the perimeter. Then the area is A = sqrt( s · (s - a) · (s - b) · (s - c) ). For sides 3, 4, 5: s = (3 + 4 + 5)/2 = 6, and A = sqrt(6 · 3 · 2 · 1) = sqrt(36) = 6. That matches the right-triangle answer (1/2) · 3 · 4 = 6, as it must.
Three positive numbers only form a real triangle if each side is shorter than the sum of the other two (the triangle inequality). If that fails, one of the factors (s - a), (s - b), (s - c) turns negative, the expression under the square root goes negative, and the formula correctly refuses to produce a real area — its way of telling you the triangle cannot exist.
A triangle with sides 13, 14, 15 has s = (13 + 14 + 15)/2 = 21, so A = sqrt(21 · 8 · 7 · 6) = sqrt(7056) = 84 square units.
Area from three sides alone — no height, no angle.
If the three lengths violate the triangle inequality, the value under the square root is negative and no triangle exists — the formula is honest about impossible side sets.