Why circular segment geometry matters
A circular segment is the area cut off a circle by a single chord, the shape you get from a curved slot, a partially cut disc, or a rounded notch on the edge of a part. Neither the area nor the depth of that shape scales in a simple, intuitive way with the angle involved, which is why this calculation trips people up when eyeballed.
How the numbers are calculated
Segment area equals one half of the radius squared, multiplied by the segment angle in radians minus the sine of that angle. Segment height, also called the sagitta, is the radius multiplied by one minus the cosine of the angle, and represents how deep the segment cuts into the circle measured from the chord to the arc. Both formulas require the angle in radians, not degrees, which is a common source of errors when working this out by hand.
Practical tips
The sagitta value is especially useful for verifying a curved cut with a height gauge or CMM, since it tells you exactly how far the deepest point of the arc sits from a straight edge laid across the opening. For small angles, the segment height grows slowly at first, so do not assume a shallow-looking arc has a proportionally shallow depth without checking the actual number.