Which formula can be used to calculate the area of a triangle when given the lengths of all three sides?

The correct formula for calculating the area of a triangle when the lengths of all three sides are known is ( A = \sqrt{s(s-a)(s-b)(s-c)} ). This formula is known as Heron's formula.

In this formula, ( s ) represents the semi-perimeter of the triangle, which is calculated as ( s = \frac{(a+b+c)}{2} ), where ( a, b, ) and ( c ) are the lengths of the sides of the triangle. Heron's formula is advantageous because it does not require knowledge of the triangle's height or the angles, making it universally applicable for any triangle as long as the side lengths are known.

To derive the area using this method, you first compute the semi-perimeter and then use it in the formula along with the three side lengths. This allows you to directly solve for the area without needing additional measurements, like height.

The other formulas provided are specific to certain conditions or configurations of triangles. The first formula, which is ( A = \frac{1}{2} bl ), is used to calculate the area of a triangle when the base and height are known, which is not applicable when only the side lengths are given.