Area of a Triangle and Parallelogram Using Vectors

The idea

The magnitude of the cross product of two vectors equals the area of the parallelogram they form as adjacent sides. A triangle is just half of that parallelogram, so it uses the same cross product, halved:

\text{Area of parallelogram} = |\vec{a} \times \vec{b}|, \qquad \text{Area of triangle} = \dfrac{1}{2}|\vec{a} \times \vec{b}|

Worked example: a triangle from three points

Find the area of the triangle with vertices $A(1,1,1)$ , $B(1,2,3)$ , and $C(2,3,1)$ .

Solution: Build two side vectors from the same corner, $A$ :

\overrightarrow{AB} = (0,1,2), \qquad \overrightarrow{AC} = (1,2,0)

\overrightarrow{AB} \times \overrightarrow{AC} = (1(0)-2(2),\; 2(1)-0(0),\; 0(2)-1(1)) = (-4,2,-1)

The magnitude is $\sqrt{16+4+1}=\sqrt{21}$ , so the triangle's area is:

\text{Area} = \dfrac{1}{2}\sqrt{21} = \dfrac{\sqrt{21}}{2}

Worked example: a parallelogram from its sides

Find the area of a parallelogram whose adjacent sides are $\vec{a} = \hat{i}-\hat{j}+3\hat{k}$ and $\vec{b} = 2\hat{i}-7\hat{j}+\hat{k}$ .

Solution:

\vec{a}\times\vec{b} = ((-1)(1)-(3)(-7),\; (3)(2)-(1)(1),\; (1)(-7)-(-1)(2)) = (20,5,-5)

The magnitude gives the area directly:

|\vec{a}\times\vec{b}| = \sqrt{400+25+25} = \sqrt{450} = 15\sqrt{2}

Area of a Triangle and Parallelogram Using Vectors

The idea

Worked example: a triangle from three points

Worked example: a parallelogram from its sides

Continue learning

Area of a Triangle and Parallelogram Using Vectors

The idea

Worked example: a triangle from three points

Worked example: a parallelogram from its sides

Continue learning