# How to calculate area of triangle having its points 2D coordinates?

We have points A, B & C in 2D plane. How having point coordinates $(x, y)$ to calculate area of triangle formed by them?

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English, please. – Zarrax Nov 16 '10 at 3:58

To make Rahul's comment more explicit, the determinant formula

$$\frac12 \begin{vmatrix}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{vmatrix}$$

where the $(x_i,y_i)$ are the coordinates of the corners, gives the (signed) area of the triangle. For a guaranteed positive result, the points are to be taken anticlockwise.

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Straying slightly off-topic, I never noticed before I saw your answer that the obvious way to write the absolute value of a determinant is pretty ugly: $\big\lvert\lvert A\rvert\big\rvert$... (Of course one may write $\lvert\det A\rvert$, but one needs foresight for that!) – Rahul Nov 16 '10 at 6:41
@Rahul: Yeah, the couple of times I had to use both determinants and norms, I always used $|\det\mathbf{A}|$. – J. M. Nov 16 '10 at 7:02

If by square you mean the area, Heron's formula is your friend. Just calculate the side lengths and the semiperimeter.

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If the coordinates of the points are known, Heron's formula is overkill. The area is just half the cross product of two edges. – Rahul Nov 16 '10 at 3:53
@Rahul: Right you are. – Ross Millikan Nov 16 '10 at 5:14