in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle. Using trigonometry The height of a triangle can be found through the application of trigonometry. Knowing SAS: Using the labels in the image on the right, the altitude is . Substituting this in the formula derived above, the area of the triangle can be expressed as: (where α is the interior angle at A, β is the interior angle at B, is the interior angle at C and c is the line AB). Furthermore, since sin α = sin (π − α) = sin (β + ), and similarly for the other two angles: Knowing AAS: and analogously if the known side is a or c. Knowing ASA: and analogously if the known side is b or c. Heron's formula The shape of the triangle is determined by the lengths of the sides. Therefore, the area can also be derived from the lengths of the sides. By Heron's formula: where is the semiperimeter, or half of the triangle's perimeter. Three other equivalent ways of writing Heron's formula are Using vectors The area of a parallelogram embedded in a three-dimensional Euclidean space can be calculated using vectors. Let vectors AB and AC point respectively from A to B and from A to C. The area of parallelogram ABDC is then which is the magnitude of the cross product of vectors AB and AC. The area of triangle ABC is half of this, The area of triangle ABC can also be expressed in terms of dot products as follows: In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to (x1,y1) and AC as (x2,y2), this can be rewritten as: Using coordinates If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by and , then the area can be computed as times the absolute value of the determinant For three general vertices, the equation is: which can be written as If the points are labeled sequentially in the counterclockwise direction, the above determinant expressions are positive and the absolute value signs can be omitted. The above formula is known as the shoelace formula or the surveyor's formula. If we locate the vertices in the complex plane