Central angle

The size of a central angle θ is 0° < θ < 360° or 0 < θ < 2π (radians). When defining or drawing a central angle, in addition to specifying the points A and B, 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 A to point B is clockwise or counterclockwise.

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.

If the central angle θ is subtended by L, then $${\displaystyle 0^{\circ }<\theta <{180}^{\circ },\theta ={\left(\frac{180L}{\pi R}\right)}^{\circ }=\frac{L}{R}.}$$

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: $${\displaystyle L=\frac{\theta }{{360}^{\circ }}\cdot 2\pi R\Rightarrow \theta ={\left(\frac{180L}{\pi R}\right)}^{\circ }.}$$

Proof (for radians)

The circumference of a circle with radius R is 2πR, and the minor arc L is the ⁠θ/2π⁠ proportional part of the whole circumference (see arc). So $${\displaystyle L=\frac{\theta }{2\pi }\cdot 2\pi R\Rightarrow \theta =\frac{L}{R}.}$$

If the central angle θ is not subtended by the minor arc L, then θ is a reflex angle and $${\displaystyle {180}^{\circ }<\theta <{360}^{\circ },\theta ={\left(360-\frac{180L}{\pi R}\right)}^{\circ }=2\pi -\frac{L}{R}.}$$

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