Equilateral triangle terms of side length "a" can be derived directly using the Pythagorean theorem or using trigonometry. The area of a triangle is half of one side "a" times the height "h" from that side: The legs of either right triangle formed by an altitude of the equilateral triangle are half of the base "a", and the hypotenuse is the side "a" of the equilateral triangle. The height of an equilateral triangle can be found using the Pythagorean theorem so that Substituting "h" into the area formula (1/2)"ah" gives the area formula for the equilateral triangle: Using trigonometry, the area of
Triangle inequality that the longest side length is less than the semiperimeter. A mathematically equivalent formulation is that the area of a triangle with sides "a", "b", "c" must be a real number greater than zero. 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