Circle
<p>A <b>circle</b> is a simple closed <a href="https://en.wikipedia.org/wiki/Shape" title="Shape">shape</a> in <a href="https://en.wikipedia.org/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a>. It is the set of all <a href="https://en.wikipedia.org/wiki/Point%28geometry%29" title="Point (geometry)">points</a> in a <a href="https://en.wikipedia.org/wiki/Plane%28mathematics%29" class="mw-redirect" title="Plane (mathematics)">plane</a> that are at a given distance from a given point, the <a href="https://en.wikipedia.org/wiki/Centre_%28geometry%29" title="Centre (geometry)">centre</a>; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the <a href="https://en.wikipedia.org/wiki/Radius" title="Radius">radius</a>.
A circle is a simple closed <a href="https://en.wikipedia.org/wiki/Curve" title="Curve">curve</a> which divides the plane into two regions: an <a href="https://en.wikipedia.org/wiki/Interior%28topology%29" title="Interior (topology)">interior</a> and an exterior. In everyday use, the term "circle" may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is only the boundary and the whole figure is called a <a href="https://en.wikipedia.org/wiki/Disk%28mathematics%29" title="Disk (mathematics)">disc</a>.
A circle may also be defined as a special kind of <a href="https://en.wikipedia.org/wiki/Ellipse" title="Ellipse">ellipse</a> in which the two <a href="https://en.wikipedia.org/wiki/Focus%28geometry%29" title="Focus (geometry)">foci</a> are coincident and the <a href="https://en.wikipedia.org/wiki/Eccentricity%28mathematics%29" title="Eccentricity (mathematics)">eccentricity</a> is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using <a href="https://en.wikipedia.org/wiki/Calculus_of_variations" title="Calculus of variations">calculus of variations</a>.</p>
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<a href="https://en.wikipedia.org/wiki/Squaring_the_circle" title="Squaring the circle">Squaring the circle</a> is the problem, proposed by <a href="https://en.wikipedia.org/wiki/Classical_antiquity" title="Classical antiquity">ancient</a> <a href="https://en.wikipedia.org/wiki/Geometers" class="mw-redirect" title="Geometers">geometers</a>, of constructing a square with the same area as a given circle by using only a finite number of steps with <a href="https://en.wikipedia.org/wiki/Compass_and_straightedge" class="mw-redirect" title="Compass and straightedge">compass and straightedge</a>.
In 1882, the task was proven to be impossible, as a consequence of the <a href="https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem" title="Lindemann–Weierstrass theorem">Lindemann–Weierstrass theorem</a> which proves that <a href="https://en.wikipedia.org/wiki/Pi" title="Pi">pi</a> (π) is a <a href="https://en.wikipedia.org/wiki/Transcendental_number" title="Transcendental number">transcendental number</a>, rather than an <a href="https://en.wikipedia.org/wiki/Algebraic_number" title="Algebraic number">algebraic irrational number</a>; that is, it is not the <a href="https://en.wikipedia.org/wiki/Root_of_a_function" class="mw-redirect" title="Root of a function">root</a> of any <a href="https://en.wikipedia.org/wiki/Polynomial" title="Polynomial">polynomial</a> with <a href="https://en.wikipedia.org/wiki/Rational_number" title="Rational number">rational</a> coefficients.</p>