What is a circle segment?

A circle segment is the region between a chord and the arc it subtends. It is different from a sector, which is the pie-slice region between two radii. The segment area equals the sector area minus the triangle area. A semicircle is the special case where the central angle is 180 degrees, making the chord a diameter.

What is the sagitta of a segment?

The sagitta (from the Latin for arrow) is the perpendicular distance from the midpoint of the chord to the arc. It tells you how deep the segment is. For a central angle theta and radius r, the sagitta equals r minus r times cos(theta divided by 2), or r times (1 minus cos(theta/2)).

What is the formula for a circle segment area?

For a segment with radius r and central angle theta in radians: area = (r squared times (theta minus sin(theta))) divided by 2. In degrees, convert theta to radians first by multiplying by pi divided by 180. The formula subtracts the triangle area (r squared times sin(theta) divided by 2) from the sector area (r squared times theta divided by 2).

Circle Segment Calculator

A circle segment is the region enclosed between a chord and the arc that it subtends. Draw any straight line across a circle and you create two segments: the minor segment (the smaller region) and the major segment (the larger region). Segments appear in structural engineering when calculating the cross-sectional area of partially filled pipes and tanks, in optics for lens design, in geography when computing the area of a spherical cap, and in woodworking when laying out curved profiles. This calculator focuses on the minor segment defined by a given central angle. You enter the radius of the circle and the central angle subtended by the chord in degrees, and the calculator returns four measurements: the segment area, the arc length (the curved boundary), the chord length (the straight boundary), and the sagitta, which is the perpendicular height of the segment from the chord to the arc. These four quantities together fully characterise the segment shape. At 180 degrees the segment becomes a semicircle. At 360 degrees it equals the full circle. The default uses r = 10 and theta = 90 degrees, giving segment area = 28.54, arc = 15.71, chord = 14.14, and sagitta = 2.93.

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Radius (r)

units

Central angle (θ)

deg

28.54

segment area (square units)

Arc length15.71

Chord length14.14

Sagitta2.93

Sector area78.54

Segment area = sector area minus triangle area. At 180 degrees the segment is a semicircle. Angle must be between 0 and 360 degrees exclusive.

How it works

For a circle of radius r and central angle theta in degrees, first convert to radians: t = theta * pi / 180. Arc length = r * t. Chord length = 2r * sin(t/2). Sector area = r^2 * t / 2. Triangle area = r^2 * sin(t) / 2. Segment area = sector area minus triangle area = (r^2 / 2) * (t - sin(t)). Sagitta = r * (1 - cos(t/2)), the perpendicular height from the midpoint of the chord to the arc.

Worked example

For r = 10, theta = 90 degrees: t = pi/2 = 1.5708 rad. Arc = 10 * 1.5708 = 15.71. Chord = 2 * 10 * sin(45) = 20 * 0.7071 = 14.14. Sector area = 100 * pi/2 / 2 = 78.54. Triangle area = 100 * sin(90) / 2 = 50. Segment area = 78.54 - 50 = 28.54. Sagitta = 10 * (1 - cos(45)) = 10 * (1 - 0.7071) = 2.93. These match the defaults pre-filled above.

Related calculators

Annulus Calculator: area between two concentric circles.
Hemisphere Calculator: volume and surface area of a hemisphere.
Parallelogram Calculator: area and diagonals of a parallelogram.
Rhombus Calculator: area and diagonals of a rhombus.

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