Detailed definition
Understanding Central Angle
A central angle is an angle whose vertex is at the center of the circle. Its sides are radii, so the angle opens from the point that defines the whole circle.
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.
Key facts
Important ideas to remember
- 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.
Where it is used
Where central angle shows up
- 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.
Common mistakes
What to watch out for
- 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.