Central angle

In geometry, a central angle is an angle whose apex (vertex) is the center of a circle and whose legs (sides) are radii intersecting the circle in two distinct points and . Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called spherical distance. The size of a central angle is or (radians). When defining or drawing a central angle, in addition to specifying the points and , one must specify whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°). Equivalently, one must specify whether the movement from point to point is clockwise or counterclockwise.

In geometry, a central angle is an angle whose apex (vertex) is the center of a circle and whose legs (sides) are radii intersecting the circle in two distinct points and . Central angles are subtended by an arc between those two points, and the arc length is the central angle of a circle of radius one (measured in radians). The central angle is also known as the arc's angular distance. The arc length spanned by a central angle on a sphere is called spherical distance. The size of a central angle is or (radians). When defining or drawing a central angle, in addition to specifying the points and , one must specify whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°). Equivalently, one must specify whether the movement from point to point is clockwise or counterclockwise. ==Formulas== If the intersection points and of the legs of the angle with the circle form a diameter, then is a straight angle. (In radians, .) Let be the minor arc of the circle between points and , and let be the radius of the circle. If the central angle is subtended by , then 0^{\circ} {{math proof|title=Proof (for degrees)|proof= The circumference of a circle with radius is , and the minor arc is the proportional part of the whole circumference (see arc). So: L=\frac{\theta}{360^{\circ}} \cdot 2 \pi R \quad \Rightarrow \quad \theta = \left( {\frac{180L}{\pi R}} \right) ^{\circ}. }} If the central angle is not subtended by the minor arc , then is a reflex angle and 180^{\circ} If a tangent at and a tangent at intersect at the exterior point , then denoting the center as , the angles (convex) and are supplementary (sum to 180°). ==Central angle of a regular polygon== A regular polygon with sides has a circumscribed circle upon which all its vertices lie, and the center of the circle is also the center of the polygon. The central angle of the regular polygon is formed at the center by the circumradii to two adjacent vertices. The measure of this angle is 2\pi/n. ==References== ==External links== interactive interactive Inscribed and Central Angles in a Circle

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