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  "body": "\n## Summary\n\nThe area of a [[triangle|0CTNFGYMM2]] is the measure of the region of the plane that the triangle bounds. It is half the product of a base and the height on that base: $T = \\frac{1}{2} b h$.\nHere $b$ is the length of a side, and $h$ is the distance from the opposite vertex to the line of that side.\nIf two sides have the lengths $a$ and $b$ and the angle $\\gamma$ between them, then $T = \\frac{1}{2} a b \\sin \\gamma$.\nFor the vertices $A = (x_A, y_A)$, $B = (x_B, y_B)$ and $C = (x_C, y_C)$ in $\\mathbb{R}^2$:\n$$T = \\tfrac{1}{2} \\left| (x_B - x_A)(y_C - y_A) - (x_C - x_A)(y_B - y_A) \\right|.$$\n\n## Remarks\n\n[[Heron's formula|0FRBZ2X4HJ]] gives the area from the three side lengths alone.\nIn $\\mathbb{R}^3$, the area of the triangle $ABC$ is $\\frac{1}{2} \\|(B - A) \\times (C - A)\\|$, with the [[cross product|0N5N15HWZ7]].\nThe coordinate formula without the absolute value is the signed area. It is positive when $A$, $B$, $C$ go round anticlockwise.\n\n## Explanations\n\n### L1 human\n\nA triangle is half of a parallelogram: two copies of it fit together into one. The parallelogram has the area base times height. So the triangle has half of that, $\\frac{1}{2} b h$.\n",
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