sector and segment area
Cut a slice of pie and you have a sector — the wedge-shaped region bounded by two radii and the arc between them. Cut a circle with a single straight chord and the smaller piece you slice off, bounded by the chord and an arc, is a segment. Both are fractions of the whole disk, and the angle at the centre tells you what fraction.
A sector is just the slice of the disk corresponding to its central angle. If the central angle is theta degrees, the sector is theta/360 of the full circle, so its area is (theta/360) · pi · r^2. In radians the formula is cleaner still: area = (1/2) · r^2 · theta. The arc bounding the sector has length r · theta (radians) or (theta/360) · 2 · pi · r (degrees).
A segment is what is left after you cut the triangle off the sector. Take the sector, then remove the triangle formed by the two radii and the chord: that triangle has area (1/2) · r^2 · sin(theta). So the segment area is (1/2) · r^2 · theta minus (1/2) · r^2 · sin(theta) = (1/2) · r^2 · (theta - sin theta), with theta in radians. The honest subtlety: subtract the triangle's area, not the chord's length — sector minus triangle, never sector minus chord.
A sector of radius 6 with central angle 60 degrees has area (60/360) · pi · 6^2 = (1/6)·36·pi = 6·pi ≈ 18.8 square units.
Sector = fraction of the disk; segment = sector minus its triangle.
Match your formula to your angle unit: (theta/360)·pi·r^2 for degrees, (1/2)·r^2·theta for radians; mixing them is a classic error, and a segment is the sector minus the triangle, not minus the chord.