Circular Segment Area Calculator
From a radius and a central angle, get the segment's area, arc length, chord, and sagitta — the four numbers that describe the slice cut off by a chord.
Two inputs, four answers
Enter the radius and the central angle in degrees and the calculator returns the area (½r²(θ − sin θ)), the arc length (rθ), the chord (2r·sin(θ/2)), and the sagitta (r(1 − cos(θ/2))) at once.
Angle in degrees, lengths share a unit
Enter the angle in degrees (0–360); the radius and the arc, chord, and sagitta share one length unit, and the area comes back in square units of it.
What is a circular segment area calculator?
Radius and angle in, the full segment out
A circular segment is the region of a circle cut off by a straight line (a chord) — the slice between that chord and the arc above it, like the rounded top of a sun rising over the horizon or the cross-section of liquid in a horizontal pipe. This calculator turns two measurements — the circle's radius r and the central angle θ that the segment spans — into the four numbers that describe it: the area (the slice itself), the arc length (the curved edge), the chord length (the straight edge), and the sagitta (the height from the middle of the chord up to the arc). Each one is fixed once you know r and θ, so those two inputs are all you need for pipe-fill problems, arch and window design, tank dipsticks, and geometry homework.
Enter the radius and the central angle in degrees to get the segment's area, arc length, chord, and sagitta instantly.
Four short formulas, all built from the radius and the central angle θ. The angle is converted to radians first (θ in radians = degrees × π / 180).
area = ½ × r² × (θ − sin θ)The area subtracts the triangle (½ r² sin θ) from the pie-slice sector (½ r² θ), leaving just the segment. The arc length is r × θ, the chord is 2 × r × sin(θ/2), and the sagitta — the bulge height — is r × (1 − cos(θ/2)). Every formula uses θ in radians, which is why the degrees you type are converted first.
Suppose you have a segment with a radius of 5 and a central angle of 90°.
Convert the angle
θ = 90° × π / 180 = π/2 ≈ 1.570796 radians.
Arc, chord, and sagitta
arc = 5 × 1.570796 ≈ 7.853982; chord = 2 × 5 × sin(45°) ≈ 7.071068; sagitta = 5 × (1 − cos(45°)) ≈ 1.464466.
Area
½ × 5² × (1.570796 − 1) ≈ 7.134955 square units — the slice itself.
The four outputs each answer a different practical question. The area (about 7.134955 square units for r = 5, θ = 90°) is the size of the slice — the wetted cross-section of a partly filled pipe, the glass in an arched window, the material in a curved cut. The arc length (about 7.853982) is the curved edge you would measure along the rim, useful for trim, weather-stripping, or the length of band needed to wrap the curve. The chord (about 7.071068) is the straight edge across the bottom of the slice — the flat span of an arch or the width of the liquid surface in a pipe. The sagitta (about 1.464466), also called the segment height or "rise", is how far the arc bulges above the middle of the chord; it is the single most useful number for measuring a curve in the field, because you can find a circle's radius from just a chord and its sagitta without ever reaching the centre. Notice the sagitta is small here compared with the radius: a 90° segment rises less than a third of the radius. As the angle grows toward 180° the segment becomes a full half-circle, the chord becomes the diameter, and the sagitta equals the radius.
The formulas are exact, but a couple of practical points are worth keeping in mind.
Degrees, the minor segment, and consistent length units
The angle here is the central angle in degrees, between 0 and 360 — not the arc length and not radians (the calculator converts to radians internally). The result is the minor segment for angles under 180° and the major segment for angles over 180°; for exactly 360° you would have the whole circle. These are the standard formulas for a true circular arc, so an ellipse, a parabola, or a hand-drawn curve will differ. The radius and the length outputs also share one unit, so keep the radius in a single length unit: a radius in centimetres gives an arc, chord, and sagitta in centimetres and an area in square centimetres, never a mix.