Calculate the area of a circular segment from radius and central angle. Includes chord length, arc length, sagitta, sector area, and minor/major segment toggle.
A circular segment is the region between a chord and the arc it subtends. Imagine slicing a pizza with a straight cut that does not pass through the center — the smaller piece is a minor segment and the larger piece is a major segment. Segments arise constantly in engineering (pipe cross-sections partially filled with fluid), architecture (arched windows), and mathematics (integration of circular regions).
The area of a circular segment is derived from two simpler quantities: the sector area and the triangle area. For a central angle θ (in radians) and radius r, the sector area is ½r²θ and the isosceles triangle formed by the two radii and the chord has area ½r²sin(θ). Subtracting gives the minor segment area: A = ½r²(θ − sin θ). The major segment is the rest of the circle: πr² − A.
Related measurements include the chord length c = 2r sin(θ/2), the arc length s = rθ, and the sagitta (the height of the segment) h = r(1 − cos(θ/2)). The sagitta is especially useful in optics for lens grinding and in civil engineering for road curve design.
This calculator lets you enter the radius and central angle in degrees or radians, toggle between minor and major segment, and instantly see all associated measurements: segment area, chord, arc, sagitta, sector area, triangle area, and the ratio of the segment to the full circle. A visual bar chart compares these areas, and a reference table shows standard ratios for common angles.
Circular segments appear whenever a straight cut meets a curved boundary. That happens in partially filled pipes, arched glazing, circular reservoirs, lens shaping, and road-curve calculations. In those settings you usually need more than just one area value. You also need the chord, arc length, and sagitta so you can translate the geometry into a fabrication dimension, a fluid depth estimate, or a structural detail.
This calculator is useful because it separates minor and major segments clearly and shows how each relates to the whole circle and the parent sector. That makes it easier to avoid a common mistake: using the right radius and angle but reporting the wrong region. The side-by-side outputs help you confirm which portion of the circle you actually mean before you use the number in design or analysis.
Minor Segment Area: A = ½r²(θ − sin θ) Major Segment Area: πr² − ½r²(θ − sin θ) Chord Length: c = 2r·sin(θ/2) Arc Length: s = r·θ Sagitta: h = r(1 − cos(θ/2)) Sector Area: ½r²θ Triangle Area: ½r²·sin θ
Result: Segment area ≈ 54.57 cm², chord ≈ 20.78 cm, arc ≈ 25.13 cm, sagitta = 6 cm
For a minor segment with radius 12 cm and angle 120°, the sector area is $ rac{1}{2} imes 12^2 imes 2.0944 approx 150.80$ cm². The triangle inside that sector has area $ rac{1}{2} imes 12^2 imes sin 120° approx 96.23$ cm², so the segment area is about 54.57 cm². The chord is $2rsin( heta/2) = 24sin 60° approx 20.78$ cm, the arc is $r heta approx 25.13$ cm, and the sagitta is $12(1 - cos 60°) = 6$ cm.
The cleanest way to understand a circular segment is to see it as a leftover region. Start with the sector defined by the radius and central angle, then remove the isosceles triangle formed by the two radii and the chord. That subtraction is why segment formulas often look more complex than sector formulas. The geometry is not harder; it simply combines two familiar pieces into one curved region.
For the same chord, a circle contains two complementary segments: the smaller minor segment and the larger major segment. In practical work, confusing those two can produce a large error because one may represent a shallow fluid depth while the other represents almost the entire cross-section. This calculator makes the distinction explicit and reports each area relative to the full circle, which helps with interpretation before you export the number into another calculation.
Hydraulics uses segment area to estimate flow in partially filled pipes and culverts. Architecture uses it to size arches, lunettes, and circular window cutouts. Optics and machining use the sagitta to describe how far a curved surface rises above a chord. Because the calculator returns the chord, arc, sagitta, and region area together, it works well for both classroom geometry and applied measurement tasks.
A circular segment is the region between a chord of a circle and the arc that the chord subtends. The minor segment is the smaller region; the major segment is the larger one.
A sector is a "pie slice" bounded by two radii and an arc. A segment is bounded by a chord and an arc — it is the sector minus the triangle formed by the two radii.
The sagitta (or versine height) is the perpendicular distance from the midpoint of the chord to the arc. Formula: h = r(1 − cos(θ/2)).
First compute θ = 2·arcsin(c / 2r), then use A = ½r²(θ − sin θ). This calculator accepts the angle directly.
Segments appear in fluid flow through partially filled pipes, arch and dome design, lens optics, road curve superelevation, and computing cross-sectional areas in hydraulics. Use this as a practical reminder before finalizing the result.
Yes — when the central angle exceeds 180°, the "minor" segment is actually larger than a semicircle. You can also toggle to major segment mode for the complementary region.