Circular Segment Calculator

How it works

A circular segment is the region cut off a circle by a straight chord: the shape of liquid lying in the bottom of a horizontal pipe, or the curved top of an arched window. It is the sector between the two radii to the chord's ends minus the triangle those radii make with the chord.

Four measurements describe it: the circle's radius r, the chord c, the height h (also called the sagitta, from the middle of the chord to the arc), and the central angle θ. Any two fix the other two. From the radius and height, cos(θ ÷ 2) = 1 − h ÷ r; from the chord and height, the intersecting-chords theorem gives r = (c² + 4h²) ÷ (8h); from the radius and chord, sin(θ ÷ 2) = c ÷ (2r).

Once r and θ are known, the area is ½ r² (θ − sin θ) with θ in radians, the arc is r θ, the chord is 2r sin(θ ÷ 2) and the height is r (1 − cos(θ ÷ 2)). A height greater than the radius means the segment is the larger part of the circle, and the formulas handle that automatically; a height of exactly twice the radius is the whole disc.

Formula

θ from r and h:  θ = 2 × acos(1 − h ÷ r)
θ from r and c:  θ = 2 × asin(c ÷ (2r))      (minor segment)
r from c and h:  r = (c² + 4h²) ÷ (8h)
r from c and θ:  r = c ÷ (2 × sin(θ ÷ 2))
r from h and θ:  r = h ÷ (1 − cos(θ ÷ 2))
area   = ½ × r² × (θ − sin θ)      (θ in radians)
arc    = r × θ
chord  = 2 × r × sin(θ ÷ 2)
height = r × (1 − cos(θ ÷ 2))

Example

Liquid 2 ft deep in a horizontal tank of 5 ft radius: cos(θ ÷ 2) = 1 − 2 ÷ 5 = 0.6, so θ ÷ 2 = 53.13° and θ ≈ 106.26° (1.8546 rad). The chord is 2 × √(2 × 5 × 2 − 4) = 8 ft, the arc is 5 × 1.8546 ≈ 9.27 ft, and the wetted cross-section is ½ × 25 × (1.8546 − 0.96) ≈ 11.18 ft².

Measuring the chord and height of an arch instead, 8 ft across and 2 ft high, gives r = (64 + 16) ÷ 16 = 5 ft and the same segment.

Assumptions and limitations

• From the radius and chord, the calculator returns the minor segment (central angle at most 180°). A chord cuts a circle into two segments; for the larger one, enter the radius and the larger height instead.
• The central angle is entered in degrees, between 0° and 360°; 360° is the full disc, which has no chord, so it is not allowed when solving from the chord.
• All lengths are in the unit you choose; the unit labels the result and does not convert anything. The area is in that unit squared.
• Results are rounded for display only.

Frequently asked questions