Arcs & Central Angles

A central angle has its vertex at the centre. The arc it cuts off has exactly the same degree measure. A 70° central angle intercepts a 70° arc.

Arcs come in sizes:

• Minor arc: less than 180°, named with two letters.
• Major arc: more than 180°, named with three letters so nobody confuses it with the minor one.
• Semicircle: exactly 180°, cut off by a diameter.

Arcs on the same circle add: adjacent arcs combine into a bigger arc, and a full circle is 360°.

Do not confuse arc *measure* with arc *length*. Measure is in degrees and does not care how big the circle is. Length is an actual distance and does. Two circles of different sizes can both have a 60° arc with completely different lengths.

A central angle's measure equals the measure of its intercepted arc, which is the definition of arc measure rather than a theorem.

Arc addition: adjacent arcs sum, and the arcs of a full circle total 360°. A major arc's measure is 360° − the corresponding minor arc.

The distinction between arc measure (degrees, scale-invariant) and arc length (a distance, proportional to r) is the conceptual crux, and it is what makes radian measure natural: dividing arc length by radius gives a scale-free number, θ = s/r.

In congruent circles, or within the same circle, congruent central angles intercept congruent arcs and congruent chords — all three congruences are equivalent, which is why chord problems can be converted into arc problems and back.

The three-letter naming convention for major arcs exists because two letters alone are ambiguous: the same two endpoints bound both a minor and a major arc.