Cirklen

Cirklen is the definite form of cirkel in Danish, and is commonly used to refer to the geometric circle as well as to round shapes in everyday language. In mathematics, the circle is defined as the set of all points in a plane that are at a fixed distance, called the radius, from a given point called the center. This set forms a one-dimensional boundary that encloses a disc.

If the center is at coordinates (h, k) and the radius is r, the circle is described by the equation (x − h)² + (y − k)² = r² in Cartesian coordinates. The diameter of the circle is d = 2r, and the circumference, or boundary length, is C = 2πr. The area enclosed by the circle is A = πr². A unit circle has r = 1, and thus C = 2π and A = π.

In trigonometry, the unit circle (radius 1) is a central concept, linking angles to coordinates on the circle through sine and cosine. Any central angle θ (in radians) subtends an arc length s = rθ. Circles also connect to other geometric ideas, such as chords, tangents and arcs, and they serve as a fundamental reference shape in geometry, design, and various applied fields.

In Danish usage, the term cirklen appears in mathematical texts, education, and descriptions of round objects, and it forms part of phrases like cirkelens centrum (the circle’s center) or cirkelbue (arc of a circle). The circle is a basic object with well-established properties and wide-ranging applications.