What is a circular segment?
A circular segment is the region cut from a circle by a chord: the straight line joins two points on the circumference, while the curved boundary is the corresponding arc. Unlike a sector, a segment does not use two radii as its boundary.
That distinction matters. A sector is naturally described by radius + central angle; a segment is the “cap” between an arc and a chord. The key quantity called the height or sagitta is the perpendicular rise from the chord to the arc.
Sector vs segment
A sector is the wedge formed by two radii and an arc. A segment is the part of that sector left after removing the isosceles triangle between the two radii and the chord.
Why “sagitta” matters
In construction and engineering drawings, the sagitta gives the rise of a curved edge above its chord. It is one of the most useful dimensions for turning a measured chord into a circle-arc model.
Circular segment formulas
The cleanest path starts with r and θ. Trigonometric functions use radians, so a degree input is converted using θrad = θdegπ/180.
How the segment-area formula is built
The most useful geometric identity is segment = sector − triangle. For a central angle θ in radians, the sector area is ½r²θ. The triangle formed by the two radii has area ½r²sinθ. Subtracting gives ½r²(θ − sinθ).
What happens at familiar angles?
Special angles are excellent mental checks. They let a student verify whether a calculator output has the right scale before trusting the last decimal place.
| Angle | Segment type | Area as a share of circle | Geometry check |
|---|---|---|---|
60° | Minor | about 2.26% | c ≈ 1.732r |
90° | Minor | ≈ 9.06% | c ≈ 1.414r |
180° | Half-circle | 50% | c = 2r |
270° | Major | ≈ 90.94% | c ≈ 1.414r |
Circular segment examples
The examples below use the same formulas as the calculator. The point is not memorising a button press; it is seeing how radius and angle flow through the geometry.
r = 10 cm, θ = 90°
Convert 90° to π/2 radians and substitute into the area formula.
The full circle area is about 314.16 cm², so the segment uses about 9.06% of the circle.
r = 8 m, θ = 180°
At 180°, sinπ = 0, so the segment formula reduces to half the area of the circle.
The chord is the diameter: c = 16 m, and the sagitta is h = 8 m.
r = 2.5 m, θ = 40°
A narrow angle creates a shallow cap. The result is much smaller than the corresponding sector because the triangle removes most of that wedge.
A = ½r²(θ − sinθ)
This is a useful scale check: small θ should give a small segment area.
Known chord and height
If you measure chord c and sagitta h instead, the radius can be reconstructed with the standard relation.
This reverse form is useful in measured drawings and circular-arc fitting.
Where circular segments are used
Circular-segment geometry appears wherever a straight chord meets a curved edge. In engineering and architecture, the same radius, span/chord and rise relationships are used to describe arches; in optics, sagitta is a standard way to describe the depth of a curved lens surface.
Curved structural openings
An arch can be idealised as part of a circle. The chord is the span line and the rise corresponds to the vertical depth of the arc above that chord. Segment geometry provides the basic shape relationships before a full structural model is applied.
Arched windows and façades
When a curved top is specified by a radius and chord, the sagitta tells the designer how high the arch rises. That same geometry controls the shape of the opening and helps estimate material or opening area.
How to check a segment result
Check the angle
Make sure your calculator input is in degrees while the formula is evaluated in radians. Mixing the two is one of the most common sources of large errors.
Check the scale
For a fixed angle, all linear dimensions should scale with r. If you double r, the area should become four times as large.
Check the chord
The chord cannot exceed the diameter. For a 180° segment, c = 2r; for smaller angles it is less than 2r.
Check the type
Below 180° the segment is minor; at 180° it is a semicircle; above 180° it is a major segment under this direct-angle interpretation.
Segment calculator FAQ
A sector is bounded by two radii and an arc. A segment is bounded by one chord and one arc.
Yes, the standard formulas use radians for θ. This calculator accepts degrees and performs the degree-to-radian conversion internally.
The sagitta is the segment height h: the perpendicular distance from the chord to the arc.
Yes. When the defining angle is greater than 180°, the formula describes a major segment whose area is greater than half the circle.
The chord is the straight-line distance between the arc endpoints. The arc length follows the curved boundary, so it is at least as long as the chord for the same endpoints.
For a fixed central angle, chord, height and arc length double; segment area becomes four times larger.