The turn controls the arc
How central angles behave
Central angles are especially important because, in the usual minor-arc setting, the measure of a central angle matches the measure of its intercepted arc. That makes the central angle the cleanest way to read how much of the circle has been turned.
This page keeps the angle and its arc on the same board so the relationship between turn, arc, and sector can be read at a glance.
A central angle of 60 degrees intercepts one sixth of a full circle, so its minor arc also measures 60 degrees. The matching sector takes one sixth of the disk. The center is doing the organising work in all three descriptions: angle, arc, and region.
If the central angle opens to a diameter, its measure is 180 degrees and the intercepted arc is a semicircle. This gives a useful boundary case between a small sector and a full turn, and it connects central-angle language directly to diameter vocabulary.
- A central angle has its vertex at the center of the circle.
- The sides of a central angle are radii of the circle.
- For the intercepted minor arc, the central-angle measure equals the arc measure.
- Central-angle measure controls both sector area and arc length as a fraction of the full circle.
A circle angle with a clear anchor
Where central angles help
- Use central angles when finding arc measure, arc length, or sector area.
- Use them when comparing radii and intercepted arcs in theorem problems.
- Use them as the reference angle when a circle diagram is built from the center outward.
Check the vertex first
Central-angle errors to catch
- Do not confuse a central angle with an inscribed angle whose vertex lies on the circle.
- Do not forget that the vertex must be exactly at the center.
- Do not read a central angle from the wrong intercepted arc when both major and minor arcs are visible.
