Area and volume using vectors (HL)

For area and volume formulas of common 2-D shapes and 3-D solids, see midpoint, area, volume

On this page, the vectors are interchangeable in any order in each formula.

Parallelogram

A parallelogram with adjacent edges $\bm a$ and $\bm b$ , has area

\text{area of parallelogram} = \lvert\bm a \times \bm b\rvert

Since a triangle is half the area of a parallelogram, the area of a triangle is

\text{area of triangle} = \frac 12\lvert\bm a \times \bm b\rvert

Note that this is the same as

\text{area of triangle} = \frac 12 ab \sin C

A parallelepiped is 3-D shape with six parallelogram faces, and defined by 3 edges $\bm a$ , $\bm b$ , $\bm c$ meeting at a vertex.

\text{volume of parallelepiped} = \lvert(\bm a \times \bm b) \cdot \bm c\rvert

Three vectors are coplanar if $(\bm a \times \bm b) \cdot \bm c = 0$ .

The quantity $(\bm a \times \bm b) \cdot \bm c$ is known as the scalar triple product.