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How to Calculate Area of a Triangle with Vectors

Finding the area of a triangle spanned by u and v is equal to the length of the u ×v-vector divided by two. That means it is half the area of a parallelogram. If you know the angle between the two vectors, you can use this formula:

Formula

The Area of a Triangle with a Known Angle

1 2 |u ×v| = 1 2 |u| |v| sin α, α = (u,v)

1 2 |u ×v| = 1 2 |u| |v| sin α,α = (u,v)

If you have the vectors on coordinate form, you can use this formula:

Formula

The Area of a Triangle on Vector Coordinate Form

1 2 |u ×v| = 1 2 | (x1, y1, z1 ) × ( x 2, y2, z2 ) | = 1 2|(y1z2 y2z1,z1x2 z2x1, x1y2 x2y1)|

1 2 |u ×v| = 1 2 | (x1, y1, z1 ) × ( x 2, y2, z2 ) | = 1 2 | (y1z2 y2z1,z1x2 z2x1,x1y2 x2y1)|

Two vectors and the area of the triangle spanned by the vectors

Example 1

Find the area of the triangle spanned by u = (1, 3, 2) and v = (3, 2, 4).

First, you have to find the cross product of the vectors, which turns out to be (16, 2, 11). The length of this vector will be equal to the area of the parallelogram u and v spans. That means you have to divide the length by 2 to find the area of the triangle. 1 2162 + 22 + 112 = 1 2256 + 4 + 121 = 1 2381 9.8

The area of the triangle is approximately equal to 9.8.

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