A central angle has its vertex at the center of the circle. The arc intercepted by a central angle has the same degree measure as the angle. A minor arc (less than 180 degrees) and a major arc (greater than 180 degrees) together comprise the full circle (360 degrees). A semicircle is an arc of exactly 180 degrees. The Arc Addition Postulate states that adjacent arcs can be added. Central angles and arcs connect angle measurement to arc measurement.
Draw central angles and identify their intercepted arcs. Practice finding arc measures given central angles and vice versa. Introduce three-letter arc notation for major arcs. Use the Arc Addition Postulate to find unknown arc measures. Connect to pie charts and clock angles for real-world context.
From your study of circle basics, you know that a circle is defined by its center and radius, and that all points on the circle are equidistant from the center. A central angle is simply an angle whose vertex sits exactly at that center. Because the center is the "hub" of the circle, a central angle has a uniquely direct relationship with the arc it cuts off: the arc's degree measure equals the angle's degree measure, exactly.
Why is this true? Imagine the circle as a full rotation of 360°. A central angle that opens to 90° claims exactly one quarter of that full rotation — and the arc it intercepts is exactly one quarter of the circle. The fraction of the full angle equals the fraction of the full circle. This proportionality is what makes arc measure so clean: a 60° central angle cuts off a 60° arc, always, regardless of the radius. You're dividing the circle proportionally, and both the angle and the arc share the same proportion.
This leads directly to the vocabulary you need. A minor arc is the smaller of the two arcs formed by two points on the circle — it corresponds to a central angle less than 180°. The major arc is the larger piece, corresponding to a reflex angle greater than 180°. Together they sum to 360°. A semicircle is the special case where both arcs are equal — exactly 180° each, cut by a diameter. Because two-letter arc notation (arc AB) is ambiguous for major arcs (it could refer to either arc), you use three letters to specify the path: arc ACB traces from A through point C to B, leaving no ambiguity about which arc is meant.
The Arc Addition Postulate mirrors the Segment Addition Postulate you already know from angle basics. If C is a point on arc AB (between A and B), then the measure of arc AC plus the measure of arc CB equals the measure of arc AB. This additive structure lets you build up unknown arc measures from known pieces, just as you add angle measures to find totals. One important clarification before moving on: arc *measure* (in degrees) and arc *length* (in centimeters or meters) are different quantities. Two arcs can have the same 90° measure but very different lengths if they come from circles of different radii. Arc length depends on radius; arc measure does not. That distinction becomes essential when you study arc length formulas next.