Zentralwinkel

Zentralwinkel, also known in English as the central angle, is a fundamental concept in circle geometry. It is the angle subtended by an arc at the circle’s center and is formed by two radii that share the circle’s center as their vertex. The measure of a Zentralwinkel uniquely determines the size of the corresponding arc and sector.

The central angle can be expressed in degrees or radians. A full circle has 360 degrees or 2π radians. The central angle directly relates to arc length: s = r · θ, where r is the radius and θ is the central angle in radians. Equivalently, in degrees, s = (θ_deg / 360) · 2πr or s = r · (θ_rad).

Key geometric relations include the chord length and areas of sectors and triangles formed by the central angle. The chord length c subtending a Zentralwinkel θ is c = 2r sin(θ/2) with θ in radians. The area of the corresponding sector is A_sector = (1/2) r^2 θ, while the area of the triangle formed by the two radii and the chord is A_triangle = (1/2) r^2 sin θ. For any given chord, the central angle is twice the inscribed angle that subtends the same chord, a consequence of the inscribed angle theorem.

Applications of the Zentralwinkel span various areas of mathematics and applied sciences, including trigonometry, geometry, computer graphics, surveying, and cartography. It provides a concise description of arc length, area, and chord relationships within a circle.