sirkels

Circles, or sirkels in Afrikaans, are the set of all points in a plane that are at a fixed distance from a given point called the center. The fixed distance is the radius r, and the line through the center that passes from one side to the other is the diameter d, with d = 2r. All points on a circle are equidistant from the center, and a circle is a closed, perfectly symmetric curve with an infinite number of lines of symmetry.

Key measurements and properties include the circumference, the distance around the circle, given by C = 2πr, and the area, representing the region inside the circle, given by A = πr^2. A full circle corresponds to 360 degrees or 2π radians. The circle is a fundamental shape in Euclidean geometry and serves as a basis for many formulas and constructions.

Equations and representations: In Cartesian coordinates, a circle with center (h, k) and radius r is described by (x − h)^2 + (y − k)^2 = r^2. In polar coordinates, a circle with constant radius is r = constant. Important derived quantities include arc length s = rθ for a central angle θ (in radians) and the area of a sector, which is A_sector = (1/2) r^2 θ. The length of a chord subtending a central angle θ is c = 2r sin(θ/2). A tangent to a circle is perpendicular to the radius at the point of tangency.

Variants and applications: Circles can be inscribed in polygons (incircles) or circumscribed around polygons (circumcircles). They are widely used in design, engineering, clocks, and various areas of mathematics, including geometry and trigonometry.