5 ways to calculate the area of a triangle

Construct a perpendicular line segment from one vertex to the base of the triangle. Calculate the length of the base and of the height to determine the area of the triangle using .

Calculate the length of all three sides and use Heron's formula, , where s is the semiperimeter, , of the triangle. Did you get the same answer?

Calculate the measure of any angle using the Law of Cosines, , and then use the formula, derived from right triangle trigonometry, . You can use any pair of sides and their included angle. Did you get the same result?

Calculate the determinant of the matrix formed by entering the coordinates of points followed by the number 1, as in the image above, in a 3 X 3 matrix. If it is positive, multiply it by and if it is negative, by . Did you get the same result?

Construct a rectangle in which the triangle is inscribed. Calculate the areas of the right triangles outside the triangle and subtract them from the area of the rectangle.

Which of the methods above would apply to a triangle in 3-dimensions?

Find the vector that is perpendicular to the plane containing the triangle. The area of the triangle is 1/2 the magnitude of this vector.