# Calculation of Areas and Volumes

#### Area of Parallelogram

Geometrically, the magnitude of the cross product of two vectors coincides with the area of the parallelogram whose sides are those vectors.

#### Example

Given the vectors and , find the area of the parallelogram whose sides are the vectors and ·

#### Area of a Triangle

#### Example

Determine the area of the triangle whose vertices are the points A = (1, 1, 3), B = (2, −1, 5) and C = (−3, 3, 1).

### Volume of a Parallelepiped

Geometrically, the absolute value of the triple product represents the volume of the parallelepiped whose edges are the three vectors that meet in the same vertex.

Find the volume of the parallelepiped formed by the following vectors:

### Volume of a Tetrahedron

The volume of a tetrahedron is equal to 1/6 of the absolute value of the triple product.

Calculate the volume of the tetrahedron whose vertices are the points A = (3, 2, 1), B = (1, 2, 4), C = (4, 0, 3) and D = (1, 1, 7).