Английская Википедия:Central angle

Материал из Онлайн справочника
Перейти к навигацииПерейти к поиску

Шаблон:Short description

Файл:Sector central angle arc.svg
Angle AOB is a central angle

A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. 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).[1] 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 Шаблон:Math is Шаблон:Math or Шаблон:Math (radians). When defining or drawing a central angle, in addition to specifying the points Шаблон:Mvar and Шаблон:Mvar, 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 Шаблон:Mvar to point Шаблон:Mvar is clockwise or counterclockwise.

Formulas

If the intersection points Шаблон:Mvar and Шаблон:Mvar of the legs of the angle with the circle form a diameter, then Шаблон:Math is a straight angle. (In radians, Шаблон:Math.)

Let Шаблон:Math be the minor arc of the circle between points Шаблон:Mvar and Шаблон:Mvar, and let Шаблон:Mvar be the radius of the circle.[2]

Файл:Angle central convex.svg
Central angle. Convex. Is subtended by minor arc Шаблон:Math

If the central angle Шаблон:Math is subtended by Шаблон:Math, then <math display="block"> 0^{\circ} < \Theta < 180^{\circ} \, , \,\, \Theta = \left( {\frac{180L}{\pi R}} \right) ^{\circ}=\frac{L}{R}.</math>

Шаблон:Math proof \cdot 2 \pi R \, \Rightarrow \, \Theta = \left( {\frac{180L}{\pi R}} \right) ^{\circ}.</math>}}

Файл:Angle central reflex.svg
Central angle. Reflex. Is not subtended by Шаблон:Math

Шаблон:Math proof

If the central angle Шаблон:Math is not subtended by the minor arc Шаблон:Math, then Шаблон:Math is a reflex angle and <math display="block"> 180^{\circ} < \Theta < 360^{\circ} \, , \,\, \Theta = \left( 360 - \frac{180L}{\pi R} \right) ^{\circ}=2\pi-\frac{L}{R}.</math>

If a tangent at Шаблон:Math and a tangent at Шаблон:Math intersect at the exterior point Шаблон:Math, then denoting the center as Шаблон:Math, the angles Шаблон:Math (convex) and Шаблон:Math are supplementary (sum to 180°).

Central angle of a regular polygon

A regular polygon with Шаблон:Math 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 radii to two adjacent vertices. The measure of this angle is <math>2\pi/n.</math>

See also

References

Шаблон:Reflist

External links