 Heron's formula for the area is

In terms of either area expression, the triangle inequality imposed on all sides is equivalent to the condition that the expression under the square root sign be real and greater than zero (so the area expression is real and greater than zero).The triangle inequality provides two more interesting constraints for triangles whose sides are "a, b, c", where "a ≥ b ≥ c" and "formula_8" is the golden ratio, as

In the case of right triangles, the triangle inequality specializes to the statement that the hypotenuse is greater than either of the two sides and less than their sum.The second part of this theorem is already established above for any side of any triangle.