Keskuskulman

Keskuskulman, in English central angle, is the angle formed at the center of a circle by two radii drawn to the endpoints of an arc. It determines the size of the arc and the corresponding sector.

Measurement and formulas: The central angle can be expressed in radians or degrees. If the arc length is s and the circle has radius r, the central angle θ in radians equals s divided by r (θ = s/r). In degrees, θ_deg = θ × 180/π. Equivalently, the arc length is s = r × θ when θ is in radians.

Relations and properties: The chord length c subtended by a central angle θ is c = 2r sin(θ/2). The area of the sector defined by the central angle is (1/2) r^2 θ (with θ in radians). The arc length and the central angle are directly related through the radius, so the central angle also determines how much of the circle’s circumference is covered.

Relationships within circles: The sum of all central angles around a circle’s center is 360 degrees (2π radians). For a regular polygon with n sides inscribed in a circle, each central angle between adjacent vertices measures 360/n degrees. The inscribed angle subtending the same arc is half the central angle.

Example: A circle with radius 5 units and arc length 10 units yields θ = s/r = 10/5 = 2 radians, about 114.6 degrees. The corresponding sector area is 0.5 × 25 × 2 = 25 square units.