Cercles

Cercles, commonly referred to in English as circles, are the set of all points in a plane at a fixed distance from a given point, the center. The fixed distance is called the radius, and the diameter is twice the radius. A circle is a two‑dimensional shape with infinite symmetry: any line through the center intersects the circle at two opposite points.

Key measures are radius r, diameter d = 2r, area A = π r^2, and circumference C = 2π r. When the radius is 1, the circle is called the unit circle, a fundamental reference in trigonometry; many properties are expressed with pi, the constant equal to the ratio of a circle’s circumference to its diameter.

In analytic geometry, a circle with center (h, k) and radius r has equation (x − h)^2 + (y − k)^2 = r^2. The unit circle centered at the origin is x^2 + y^2 = 1, with parametrizations x = h + r cos t and y = k + r sin t for angle t.

Circles appear in many geometric constructions and concepts: an inscribed circle (incircle) fits inside a polygon, tangent to all its sides; a circumscribed circle (circumcircle) passes through all the polygon’s vertices; concentric circles share the same center. Arcs, sectors, and segments are portions or divisions of a circle used in geometry and design.

Historically, the circle has been a central object in mathematics since antiquity. Euclid established foundational properties, and Archimedes contributed methods to approximate π. Today, circles underpin many applications in science, engineering, navigation, and computer graphics.