# Central angle

Angle AOB forms 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 thereby subtending an arc between those two points whose angle is (by definition) equal to that of the central angle itself.[1] It is also known as the arc segment's angular distance.

When defining or drawing a central angle, in addition to specifying the points A and B, one must specify and/or denote whether the angle being defined is the convex angle (<180°) or the reflex angle (>180°).

• The size of a central angle Θ is: 0°<Θ<360° оr 0<Θ<2π (radians)

## Formulas

• If the intersection points A and B of the legs of the angle with the circle form a diameter, then Θ=180° is a straight angle. (In radians, Θ=π.)

Let L be the minor arc of the circle between points A and B, and let R be the radius of the circle.[2]

 Central angle. Convex. Includes minor arc L
• If the central angle Θ includes L, then
$0^{\circ} < \Theta < 180^{\circ} \, , \,\, \Theta = \left( {\frac{180L}{\pi R}} \right) ^{\circ}=\frac{L}{R}$


Proof (for degrees): The circumference of a circle with radius R is: 2πR, and the minor arc L is the (Θ/360°) proportional part of the whole circumference (see arc). So:

$L=\frac{\Theta}{360^{\circ}} \cdot 2 \pi R \, \Rightarrow \, \Theta = \left( {\frac{180L}{\pi R}} \right) ^{\circ}$
 Central angle. Reflex. Does not include L

Proof (for radians): The circumference of a circle with radius R is: 2πR, and the minor arc L is the (Θ/) proportional part of the whole circumference (see arc). So:

$L=\frac{\Theta}{2 \pi} \cdot 2 \pi R \, \Rightarrow \, \Theta = \frac{L}{R}$
• If the central angle Θ does not include the minor arc L, then the Θ is a reflex angle and:
$180^{\circ} < \Theta < 360^{\circ} \, , \,\, \Theta = \left( 360 - \frac{180L}{\pi R} \right) ^{\circ}=2\pi-\frac{L}{R}$

• If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠BOA (convex) and ∠BPA are supplementary (sum to 180 °).