Circular Segment Area Calculator
Enter a radius and one additional measurement to calculate the segment area, central angle, height, arc, chord, perimeter, and share of the full circle.
Known measurements
Keep the radius and choose which second measurement should drive the calculation. The remaining fields update automatically.
Changing the unit converts every current length and area value.
Trigonometric formulas use radians internally; degrees are converted automatically.
Required. Enter a non-negative decimal greater than zero, using a period as the decimal separator.
Valid from 0° to 360°, or 0 to 2π radians.
Also called sagitta. It must be between 0 and twice the radius.
It must be between 0 and the full circumference, 2πr.
It must be between 0 and the diameter, 2r. Use the advanced setting for the major branch.
It must be between 0 and the full-circle area, πr².
Advanced interpretation
A radius and chord define two complementary segments. This setting chooses the intended branch when chord length is the known measurement.
Live results
19.8168 in²
132.84°
3 in
11.5928 in
9.1652 in
20.7579 in
25.23%
This is a minor segment because its central angle is below 180°.
Calculated relationships
| Property | Symbol | Current value | Relationship |
|---|---|---|---|
| Radius | r | 5 in | Known input |
| Central angle | α | 132.84° | 2 × arccos((r – h) / r) |
| Height | h | 3 in | r × (1 – cos(α / 2)) |
| Arc length | L | 11.5928 in | r × α, with α in radians |
| Chord length | c | 9.1652 in | 2r × sin(α / 2) |
| Segment area | A | 19.8168 in² | ½r² × (α – sin α) |
| Segment perimeter | P | 20.7579 in | L + c |
| Circle share | A / πr² | 25.23% | Segment area ÷ full-circle area |
The table and downloaded workbook use the same canonical calculation model. Displayed values are rounded for readability; the XLSX stores typed numeric values at higher precision.
How to use the circular segment area calculator
What this calculator does. It reconstructs the main measurements of a circular segment – the region bounded by a chord and its corresponding arc – from a circle radius and one additional known measurement. It calculates segment area, central angle, height or sagitta, arc length, chord length, segment perimeter, and the segment's percentage of the full circle. The model covers minor, semicircular, major, and limiting zero/full-circle cases. It is a geometric identity calculator, not a tolerance, structural-capacity, material-volume, or manufacturing-fit assessment.
When to use it. Use it to estimate the liquid cross-section in a partly filled horizontal pipe, size an arched opening from a measured rise, recover arc and chord dimensions for fabrication, or check a geometry exercise involving sectors and segments. A circular segment is distinct from a sector: the segment excludes the triangle between the two radii, a relationship described in the Wolfram MathWorld circular segment reference.
How to calculate. The calculator opens with a complete demonstration: a 5-inch radius and 3-inch height. Its results and a validated example XLSX are available immediately.
- Choose the Length unit and Angle unit. Unit changes convert the current geometry; they do not change its physical size.
- Enter Radius (r), then select the Second known measurement that you actually know. The chosen field becomes editable and the other measurement fields become calculated read-only values.
- Replace the selected value, then read Segment area and the supporting live results. Use the relationships table to see the formulas behind each value.
- Select Download Excel to export the current typed values and outputs. Reset clears the demonstration and all calculated content; export stays disabled until a complete valid radius-plus-measurement pair is entered again.
Input guide. Length unit is required and accepts mm, cm, m, in, or ft; for example, “in” displays the startup values as 5 in and 3 in. Switching units converts radius, height, arc, chord, perimeter, and area consistently. Angle unit is required and accepts degrees or radians. Degrees are easier to read, while the formulas use radians internally; NIST explains that a radian is the angle subtended by an arc equal in length to the radius. Radius (r) is required, must be greater than zero, and accepts ordinary US-style decimals such as 5, 2.75, or 1,250.5; decimal commas and scientific notation are rejected to prevent silent reinterpretation. A larger radius scales all lengths proportionally and area by the square of the scale factor.
Second known measurement is required and chooses one of five drivers. Central angle (α) accepts 0 – 360° or 0 – 2π rad; for example, 132.843643° reproduces the startup shape. Height (h) accepts 0 through 2r; the 3-inch example is the segment's maximum perpendicular rise from chord to arc. Arc length (L) accepts 0 through the full circumference 2πr; longer arcs produce larger segment areas. Chord length (c) accepts 0 through the diameter 2r. Because one chord corresponds to complementary minor and major segments, use Chord interpretation to select the intended branch; the setting matters only when chord length is the driver. Segment area (A) accepts 0 through πr². The calculator numerically solves the area equation for the unique central angle. Common mistakes are entering diameter instead of radius, mixing linear and squared units, or entering an angle outside one full turn.
Output guide. Segment area is the exact area identity for the supplied geometry, displayed in squared length units. Central angle is the arc's angle at the circle center. Height is the sagitta, Arc length is the curved boundary, and Chord length is the straight boundary. Segment perimeter equals arc plus chord. Circle share is the area divided by πr²; 0% is the degenerate zero segment, 50% is a semicircle, and 100% is the full-circle limit. Segment class labels the result minor, semicircle, major, zero, or full circle. The Known pair pill confirms which two measurements currently drive the model. Every row in Calculated relationships repeats the same canonical values with its defining relationship.
Worked example. With radius r = 5 in and height h = 3 in, the angle is α = 2 × arccos((5 – 3) / 5) = 2.318558961 rad = 132.84°. The area is ½ × 5² × (2.318558961 – sin 2.318558961) = 19.8168 in². The same geometry gives an arc length of 11.5928 in, a chord length of 9.1652 in, a perimeter of 20.7579 in, and a circle share of 25.23%. Those numbers match the first-open controls, results table, and workbook checkpoints.
Formula and interpretation
The area formula subtracts the signed isosceles-triangle contribution from the circular sector. With the central angle α in radians:
A = ½r²(α – sin α)For a minor segment, sin α is positive and the triangle is subtracted from the sector. For a major segment, α is above π and sin α is negative, so the formula correctly includes the complementary region. Arc length follows L = rα, while chord length follows c = 2r sin(α/2). OpenStax's overview of angles and arcs provides additional context for degree and radian measures.
Accuracy, boundaries, and common mistakes
Calculations use full JavaScript floating-point precision and round only when values are displayed. The XLSX contains numeric cells rather than copied screen text. At zero angle, height, arc, and area are zero. At 180°, the result is a semicircle. At the full-circle limit, height reaches 2r, arc reaches 2πr, chord returns to zero, and area reaches πr². Very small values can display as zero after rounding even though the internal value remains positive.
Use one consistent length unit for measured inputs. Do not place a squared-unit value into a linear field, and do not use chord length alone without deciding whether the intended segment is minor or major. For engineering or fabrication work, separately account for measurement uncertainty, wall thickness, machining allowance, and applicable standards.