What Is a Circular Segment?
A circular segment is the region between a chord of a circle and the arc it subtends. Think of it as a "slice" of a circle cut off by a straight line (chord). Unlike a sector (which is bounded by two radii), a segment is bounded by a chord and an arc.
The area of a circular segment equals the area of the corresponding sector minus the area of the triangle formed by the two radii and the chord.
Segment Area Formulas
Segment Area (Radians)
The direct formula using the angle in radians.
Sector Area
The pie-slice area from center to arc.
Triangle Area
The isosceles triangle formed by two radii and the chord.
How to Calculate Segment Area
- Identify the radius (r) and central angle (θ).
- Convert the angle to radians if given in degrees: θ(rad) = θ(deg) × π/180.
- Calculate the sector area: A_sector = (1/2)r²θ.
- Calculate the triangle area: A_triangle = (1/2)r²sin(θ).
- Subtract: Segment Area = Sector Area - Triangle Area.
Related Measurements
Chord Length
The chord length is given by c = 2r sin(θ/2). This is the straight-line distance between the two endpoints of the arc.
Segment Height (Sagitta)
The height of the segment (sagitta) is h = r(1 - cos(θ/2)). This is the maximum distance from the chord to the arc.
Arc Length
The arc length is L = rθ (in radians). This is the curved distance along the arc of the segment.
Example Calculation
Find the segment area for r = 10, θ = 90°:
- Convert: θ = 90 × π/180 = π/2 ≈ 1.5708 rad
- Sector area = (1/2)(100)(1.5708) ≈ 78.5398
- Triangle area = (1/2)(100)(sin 1.5708) = (1/2)(100)(1) = 50
- Segment area = 78.5398 - 50 = 28.5398 sq units
Applications
- Civil engineering: calculating cross-sectional areas of tunnels and pipes.
- Architecture: designing arched windows and doorways.
- Fluid mechanics: determining flow area in partially filled pipes.
- Optics: computing lens segment areas.