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  "body": "\n## Statement\n\nThe number $\\sqrt{\\pi}$ is not a [[constructible number|0FMN2MN7TM]].\nSo the circle cannot be [[squared|0BSXRER81B]] with straightedge and compass.\n\n## Remarks\n\nThe proof: every constructible number is [[algebraic over $\\mathbb{Q}$|0SSSM1D475]]. If $\\sqrt{\\pi}$ were constructible, then $\\pi = (\\sqrt{\\pi})^2$ would be constructible and so algebraic.\nThis contradicts the [[transcendence of $\\pi$|0G0F7FGCGP]].\n\n## History\n\nThe impossibility was proved in 1882, as a consequence of Lindemann's proof that $\\pi$ is transcendental.\nIt had been known for decades that the transcendence of $\\pi$ would imply the impossibility.\n\n## Explanations\n\n### L1 human\n\nA square with the area of the unit circle has side $\\sqrt{\\pi}$.\nConstructible numbers are algebraic, and $\\pi$ is not, so this side cannot be constructed.\n",
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