{
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  "body": "\n## Statement\n\nLet $ABC$ be a [[triangle|0CTNFGYMM2]] in the Euclidean plane, with the side lengths $a$, $b$ and $c$. Let $s = \\frac{1}{2}(a + b + c)$ be its semiperimeter.\nThen the [[area|065JWMZW86]] $T$ of the triangle is\n$$T = \\sqrt{s (s - a)(s - b)(s - c)}.$$\n\n## Remarks\n\nAn equivalent form without $s$ is $16 T^2 = (a + b + c)(-a + b + c)(a - b + c)(a + b - c)$.\nFor example, the triangle with the sides $4$, $13$ and $15$ has $s = 16$ and $T = \\sqrt{16 \\cdot 12 \\cdot 3 \\cdot 1} = 24$.\nOne proof uses the [[law of cosines|060ZVZBV3E]] to get $\\cos \\gamma$ from the sides, and then $T = \\frac{1}{2} a b \\sin \\gamma$.\n\n## History\n\nThe formula carries the name of Heron of Alexandria, who gave a proof in his Metrica in the 1st century. It was possibly known earlier. Qin Jiushao published an equivalent formula in 1247.\n\n## Explanations\n\n### L1 human\n\nHeron's formula gives the area of a triangle when you know only its three sides. You do not need a height or an angle. For the right triangle with the sides $3$, $4$, $5$: $s = 6$, and $\\sqrt{6 \\cdot 3 \\cdot 2 \\cdot 1} = 6$, which is also $\\frac{1}{2} \\cdot 3 \\cdot 4$.\n",
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  "name": "Heron's formula",
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