Segment Area Calculator – Find the Area of a Circular Segment
A circular segment is the region enclosed between a chord and the arc it subtends. Unlike a sector (the pie-slice shape that includes the central triangle), a segment contains only the curved "cap" region. Calculating segment area is a common requirement in engineering, woodworking, architecture, and mathematics.
What Is a Circular Segment?
When a chord divides a circle into two parts, each part is a segment. The smaller part (when the chord is not a diameter) is the minor segmentand the larger part is the major segment. A diameter produces two equal semicircular segments. This tool handles all three cases automatically based on the central angle you provide.
Segment Area Formula
The area of a circular segment is calculated as:
A = (r² / 2) × (θ − sin θ)
where r is the radius of the circle and θ is the central angle in radians. This formula subtracts the isoceles triangle area from the sector area to isolate the segment region.
For a major segment(θ > 180°), the tool computes the minor segment area first and then subtracts it from the total circle area:
A_major = πr² − A_minor
If you enter the angle in degrees, the calculator converts it automatically:θ_rad = θ_deg × (π / 180).
Additional Outputs
Alongside segment area, the calculator provides:
- Arc Length — the length of the curved boundary:
L = r × θ - Chord Length — the straight edge connecting the two endpoints of the arc:
c = 2r × sin(θ / 2) - Segment Height (sagitta) — the perpendicular distance from the midpoint of the chord to the arc:
h = r × (1 − cos(θ / 2)) - Central Angle — displayed in degrees even if radians were entered, or derived from chord length if chord mode is used
Using Chord Length Instead of Central Angle
In many practical situations you may know the chord length but not the central angle. Switch to Chord Mode and enter the chord length; the tool derives the central angle using:
θ = 2 × arcsin(c / 2r)
The chord must satisfy 0 < c ≤ 2r (a chord cannot be longer than the diameter). If the chord equals the diameter, the result is a semicircle.
Practical Applications
Segment area calculations appear in many real-world contexts:
- Pipe flow engineering — determining the cross-sectional area of fluid flow in a partially filled circular pipe
- Arch structures — computing the face area of curved arch panels or windows
- Woodworking and CNC — calculating material removed when cutting circular arcs or curved recesses
- Land surveying — estimating areas bounded by curved boundaries and straight baselines
- Optics — computing lens segment areas for focal calculations
Segment vs Sector: Key Difference
The sector is the "pie slice" that includes the two straight radii and the arc. The segment is the sector minus the isoceles triangle formed by those two radii. When the central angle is small, the segment area is much smaller than the sector area. As the angle approaches 360°, both converge to the full circle area. Use this calculator for segments and the Sector Area Calculator for sectors.
Tips for Accurate Results
Use enough decimal places in your radius input for precise results. All units are consistent — if the radius is in centimetres, the area output is in cm², the arc length in cm, and so on. The calculator works with any consistent unit system (metric or imperial).