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  "body": "\n## Definition\n\nLet $d$ be the [[Euclidean distance|0NPPGHQNXQ]] on the [[Euclidean plane|0XW7V9ZZ05]] $\\mathbb{R}^2$. Let $c \\in \\mathbb{R}^2$ be a point and $r \u003e 0$ a real number.\nThe circle with the centre $c$ and the radius $r$ is the set\n$$\\{\\, x \\in \\mathbb{R}^2 \\;:\\; d(x, c) = r \\,\\}.$$\nWith $c = (a, b)$, it is the set of the points $(x, y)$ with $(x - a)^2 + (y - b)^2 = r^2$.\n\n## Remarks\n\nA chord is a segment between two points of the circle. A diameter is a chord through the centre, and its length is $2r$.\nThe circle has the length $2 \\pi r$ and bounds a disc of area $\\pi r^2$.\nA circle in $\\mathbb{R}^n$ is a circle in some plane of $\\mathbb{R}^n$. Mathlib's `EuclideanGeometry.Sphere` is the sphere of a centre and a radius in a metric space; in the plane it is a circle.\n\n## Explanations\n\n### L1 human\n\nA circle is what a compass draws: all the points at one fixed distance from the centre. The unit circle has the centre $(0, 0)$ and the radius $1$. It is the set of the points with $x^2 + y^2 = 1$.\n",
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