Circular Segment Area: Chords, Arcs, and the Mathematics of Partial Circles
Calculate the area of a circular segment with our free tool. Find chord length, arc length, and segment height using radius and central angle. Step-by-step solutions included.
A circular segment is one of the most elegant shapes in classical geometry — the region of a circle bounded by a chord and the arc between the chord's endpoints. Unlike the more commonly studied sector (the "pizza slice"), the segment requires a subtraction step that involves trigonometric functions, making its area formula a beautiful intersection of algebra and trigonometry. From architectural arches to fluid flow in partially filled pipes, understanding the circular segment is essential across engineering, physics, and design disciplines.
Key Takeaway
The area of a circular segment is calculated as A = (r²/2)(θ − sin θ), where r is the radius and θ is the central angle in radians. The segment area equals the sector area minus the triangle area, combining circular arc geometry with trigonometric area calculations.
1. Defining the Circular Segment
When a chord is drawn between two points on a circle, it divides the circle into two regions. The smaller region (for a chord shorter than the diameter) is called the minor segment, and the larger region is called the major segment. The chord itself forms one boundary of the segment, while the arc of the circle forms the other.
The circular segment is distinct from several related shapes: the sector is bounded by two radii and an arc (like a pizza slice), the segment is bounded by a chord and an arc (the "crust" of a pizza slice), and the triangle formed by the two radii and the chord is the triangle portion of the sector.
A = (r²θ/2) − (r² sin θ / 2)
A = (r²/2)(θ − sin θ)
2. The Segment Area Formula
The formula for the area of a circular segment is derived by subtracting the area of the triangle formed by the two radii and the chord from the area of the sector:
- Sector area: A_sector = (r²θ)/2, where θ is in radians
- Triangle area: A_triangle = (r² sin θ)/2, using the formula (1/2)ab sin C with a = b = r
- Segment area: A = A_sector − A_triangle = (r²/2)(θ − sin θ)
When the central angle θ is measured in degrees, the formula becomes A = (πr²)(θ°/360) − (r² sin θ°)/2, which the calculator converts automatically.
Special Cases
| Central Angle (θ) | Segment Type | Area Formula |
|---|---|---|
| 0° < θ < 180° | Minor segment | A = (r²/2)(θ − sin θ) |
| θ = 180° | Semicircle | A = πr²/2 |
| 180° < θ < 360° | Major segment | A = πr² − minor segment |
3. Related Measurements
The circular segment has several important measurements beyond its area:
- Chord length: c = 2r sin(θ/2) — the straight-line distance between the two endpoints
- Arc length: s = rθ — the distance along the circle between the endpoints
- Segment height (sagitta): h = r − r cos(θ/2) — the perpendicular distance from the chord midpoint to the arc
- Segment perimeter: P = c + s = 2r sin(θ/2) + rθ
4. Real-World Applications
Circular segment calculations appear in numerous practical contexts. In civil engineering, partially filled circular pipes use the segment area to calculate the cross-sectional area of fluid flow. In architecture, arches and curved structures involve segment geometry for material estimation. In manufacturing, sheet metal bending and arch fabrication require precise segment calculations. In hydrology, the segment area determines the wetted perimeter and cross-sectional area of channels with circular cross-sections.
5. Deriving the Formula from First Principles
The segment area formula can be derived using integration in polar coordinates. The sector area is the integral of (1/2)r² dθ from 0 to θ, which yields (r²θ)/2. The triangle area is (1/2)r² sin θ, derived from the cross product of the two radius vectors. The segment area is the difference between these two expressions.
Alternatively, using Cartesian coordinates with the center at the origin, the segment can be described as the region bounded above by the circle x² + y² = r² and below by the horizontal line y = r cos(θ/2). Integrating the difference between these curves from −r sin(θ/2) to r sin(θ/2) yields the same result.
6. Frequently Asked Questions
Can the segment area be negative?
No. For any valid central angle between 0 and 2π radians, the term (θ − sin θ) is always non-negative because sin θ ≤ θ for all θ ≥ 0. The segment area is always zero or positive.
How do I find the segment area from chord length and height?
First, compute the radius: r = (c² + 4h²)/(8h), where c is the chord length and h is the height. Then compute the central angle: θ = 2 arcsin(c/(2r)). Finally, apply the segment area formula.
What is the maximum segment area?
The maximum segment area occurs when θ = 2π (the entire circle), giving A = πr². However, for non-degenerate segments (θ < 2π), the area approaches but never reaches the full circle area.
Using Our Calculator
Enter the radius and central angle (in degrees) to instantly compute the segment area, arc length, chord length, segment height, and sector area. The calculator handles the degree-to-radian conversion automatically and provides step-by-step derivations.