Definition: A central angle has its vertex at the center of a circle, and its measure equals the measure of the arc between its two sides.
What is a central angle?
Its sides are two radii. The arc they cut off has the same degree measure as the angle. A full turn is 360 degrees, so the central angles of all arcs around a circle sum to 360.
How is it related to other angles?
An inscribed angle on the same arc measures half as much. Central angles also determine the sector area and arc length as fractions of the whole circle: a central angle of θ degrees takes up θ / 360 of the circle.
Example. A pie is cut into slices with central angles of 90, 120, and the rest. The third slice has 360 – 90 – 120 = 150 degrees, which is 150/360 = 5/12 of the pie. In a circle of radius 6, the arc for a 150 degree angle is (150/360)(2π)(6), about 15.7.
What errors occur?
Treating the arc length as if it were in degrees. The arc measure is an angle, while the arc length is a distance. Also, if the angle has its vertex on the circle rather than at the center, it is inscribed.