Segment Area Calculator
A circular segment is the region between a chord and the arc it cuts from a circle. Enter the radius and one other measurement, and this calculator returns the segment area, arc length, chord length, sagitta (segment height), perimeter, and the central angle. Choose your input method below.
What is a circular segment?
A circular segment is the region of a disk enclosed between a chord and the arc it subtends. Imagine slicing off a cap from a circle: the cap is the segment, the straight cut is the chord, and the curved edge is the arc. Every chord that is not a diameter creates two segments: the smaller one is called the minor segment and the larger one is the major segment. When the chord passes through the centre, both segments are equal semicircles.
The segment area formula and how to use it
The core formula is A = (r^2 / 2) x (theta - sin theta), where r is the radius and theta is the central angle in radians. The formula subtracts the isosceles triangle area (r^2 x sin(theta) / 2) from the circular sector area (r^2 x theta / 2), leaving just the segment. If you know the chord length c instead of the angle, use theta = 2 x arcsin(c / 2r) to convert first. If you know the sagitta h (the height from the chord midpoint to the arc), use theta = 2 x arccos(1 - h/r). A practical example: a circular pipe with radius 5 cm is filled to a height of 3 cm. The sagitta is 3 cm, giving theta = 2 x arccos(1 - 3/5) = 2 x arccos(0.4) = approximately 132.84 degrees (2.319 rad), and the segment area A = 0.5 x 25 x (2.319 - sin(2.319)) = approximately 19.8 cm^2.
All the dimensions of a segment
Five quantities fully describe a circular segment: radius r, central angle theta, chord length c, sagitta (segment height) h, and arc length s. They are linked by: c = 2r sin(theta/2), h = r(1 - cos(theta/2)), s = r x theta, and the area formula above. Knowing any two of these (with r always required as one of them, or two of c/h/theta) lets you derive the rest. The perimeter of the segment boundary is the arc length plus the chord length: P = s + c = r x theta + 2r sin(theta/2).
Real-world applications
Circular segment calculations appear in engineering and everyday life more often than you might expect. Hydraulic engineers calculate the cross-sectional flow area of a partially filled pipe or channel using the segment formula, then multiply by velocity to get flow rate. Architects use it when sizing arched windows or barrel-vault roofs where only the opening width and rise are known. Surveyors and land planners encounter arc-and-chord geometry when laying out curved roads, railways, and property boundaries. Woodworkers and fabricators cut circular segments from sheet material and need to know the resulting area for material cost estimates.
Segment area at common angles (r = 1)
| Central angle | Area (r=1) | % of circle | Chord / r | Arc / r |
|---|---|---|---|---|
| 30 degrees | 0.0091 | 0.29% | 0.5176 | 0.5236 |
| 45 degrees | 0.0198 | 0.63% | 0.7654 | 0.7854 |
| 60 degrees | 0.0906 | 2.88% | 1.0000 | 1.0472 |
| 90 degrees | 0.2854 | 9.09% | 1.4142 | 1.5708 |
| 120 degrees | 0.6142 | 19.55% | 1.7321 | 2.0944 |
| 150 degrees | 1.0890 | 34.66% | 1.9319 | 2.6180 |
| 180 degrees | 1.5708 | 50.00% | 2.0000 | 3.1416 |
| 210 degrees | 2.0526 | 65.34% | 1.9319 | 3.6652 |
| 240 degrees | 2.5274 | 80.45% | 1.7321 | 4.1888 |
| 270 degrees | 2.8562 | 90.91% | 1.4142 | 4.7124 |
| 300 degrees | 3.0524 | 97.12% | 1.0000 | 5.2360 |
| 330 degrees | 3.1327 | 99.71% | 0.5176 | 5.7596 |
Segment area as a fraction of the full circle area (pi * r^2) for unit radius. Multiply by r^2 to scale to any radius.
Frequently asked questions
What is the formula for the area of a circular segment?
The standard formula is A = (r^2 / 2) x (theta - sin theta), where r is the radius of the circle and theta is the central angle in radians. It works by computing the sector area (the pie-slice shape) and then subtracting the isosceles triangle formed by the two radii and the chord. If you are working in degrees, convert first: theta (rad) = theta (deg) x pi / 180.
What is the sagitta or segment height?
The sagitta (Latin for "arrow") is the perpendicular distance from the midpoint of the chord to the midpoint of the arc. It is also called the segment height or rise. The formula is h = r x (1 - cos(theta/2)), where r is the radius and theta is the central angle. For a semicircle the sagitta equals the radius; for a very thin sliver it approaches zero.
What is the difference between a segment and a sector?
A sector is the "pie slice" shape bounded by two radii and the arc between them. A segment is the region between a chord and the arc, with no radii as sides. You can think of the segment as a sector with the isosceles triangle (formed by the two radii and the chord) removed. The sector area formula is (r^2 x theta) / 2; the segment area subtracts the triangle: (r^2 / 2) x (theta - sin theta).
How do I find the segment area if I only know the chord and height?
From the chord c and segment height h you can recover the radius: r = h/2 + c^2 / (8h). Once you have r, compute the central angle: theta = 2 x arccos(1 - h/r), and then apply the standard area formula A = (r^2 / 2) x (theta - sin theta). This calculator performs all these steps automatically when you select "Radius and segment height" or any mode where the radius is an input.
What is a minor versus a major segment?
A chord that is not a diameter splits the circle into two unequal regions. The minor segment corresponds to the smaller central angle (less than 180 degrees) and the major segment to the reflex angle (greater than 180 degrees). For a diameter the two segments are equal semicircles. This calculator handles both by accepting central angles from just above 0 to just below 360 degrees.
How is the segment perimeter calculated?
The perimeter of a segment boundary consists of two parts: the straight chord and the curved arc. Chord length c = 2r sin(theta/2) and arc length s = r x theta (theta in radians). The total perimeter is P = c + s. For a full semicircle with radius 5 cm: c = 2 x 5 = 10 cm, s = 5 x pi = 15.71 cm, P = 25.71 cm.