a central angle

Stand at the exact centre of a circle and open two arms out to two points on the rim. The angle between your arms is a central angle, and it 'aims at' the slice of rim between those two points. The hands of a clock, pivoting at the centre of the dial, sweep out central angles as time passes.

Precisely, a central angle is an angle whose vertex is the centre of the circle and whose two sides are radii reaching out to the circle. Its defining feature is the clean link to the arc it cuts off: the measure of a central angle, in degrees, equals the measure of its intercepted arc. So a central angle of 60 degrees subtends a 60-degree arc; a full turn around the centre is 360 degrees, matching the whole rim; a straight (180-degree) central angle, with its two radii forming a diameter, subtends a semicircle. This identity is essentially the definition of an arc's angular measure.

Central angles are the yardstick that all other circle angles are measured against. The inscribed angle theorem, the tangent-chord angle, the angle between two chords, and the external-secant angle are all stated as fractions or combinations of the arcs — that is, of central angles. The key contrast to lock in: a central angle equals its arc, but an inscribed angle (vertex on the circle, not the centre) standing on the same arc is only half as large. Mixing up 'vertex at the centre' (central, equal to the arc) with 'vertex on the rim' (inscribed, half the arc) is the most common circle-angle error.