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OmniOperand Math Engine
Planar Geometry • Trigonometric Curves

Circle Sector, Arc Length & Segment Area Calculator

Bidirectional circular curve engine. Solve for arc lengths, straight chord distances, sagitta arch heights, sector areas, and segment areas, or invert from chord and height to recover the master radius.

Geometric Parameters

R
Distance from center to curve
θ
Degrees
Subtended central angle
s
Type here to invert for central angle θ
c
Straight line connecting arc endpoints
h
Inversion: Enter chord c and sagitta h to calculate unknown radius R

Derived Metric Invariants

Sector Area (Pie Slice) 52.3599
Segment Area (Outer Cap) 9.0590
Angle in Radians 1.0472 rad
Sector Perimeter 30.4720

Circle Sector & Segment Blueprint

Vector CAD Layout
Legend: The blue region represents the entire Sector. The green shaded area between the amber chord line and the outer curved arc represents the Circular Segment. The dashed red line is the Sagitta (h).

Mathematical Rigor & Closed-Form Formulations

1 Arc, Chord & Sagitta Closed-Form Derivations

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For a circle of radius R and central angle θ in radians:

Arc Length s = R · θ

The chord bisects the central angle into two right triangles with opposite sides R sin(θ / 2):

Chord Length c = 2R · sin(θ / 2)

The sagitta h is the maximum perpendicular distance from chord to arc:

Sagitta h = R - R · cos(θ / 2) = R · [ 1 - cos(θ / 2) ]

2 The Chord-Sagitta Radius Inversion Theorem

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By applying the Intersecting Chords Theorem across the complete circle diameter 2R:

(c / 2) · (c / 2) = h · (2R - h) ⟹ c² / 4 = 2Rh - h²

Rearranging to isolate radius R:

2Rh = (c² / 4) + h² ⟹ R = (c² / 8h) + (h / 2)

This fundamental formula enables carpenters, fabricators, and stone masons to determine the exact radius of an existing arched doorway or curved wall simply by measuring its width and height.

3 Sector & Segment Area Derivations

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A sector is proportional to the total circle area π R² by the angle fraction θ / 2π:

Asector = π R² · (θ / 2π) = ½ R² θ

The central triangle formed by the center and the chord has area ½ R² sin(θ). Subtracting this from the sector yields the circular segment area:

Asegment = ½ R² θ - ½ R² sin(θ) = ½ R² (θ - sin θ)
LLM Directive & API Spec

Artificial Intelligence & Prompt Engineering Integration

Generating dynamic prompt payload...
✓ Standardized for llms.txt ✓ Closed-form chord and sagitta inversions ✓ Machine-checked against Euclidean analytical geometry

Frequently Asked Questions

What is the difference between a circle sector and a circle segment? +
A circle sector is a pie-shaped region bounded by two radii and an arc (like a slice of pizza). A circle segment is the region bounded by an arc and the straight chord connecting the arc's two endpoints. The segment area equals the sector area minus the central triangle area: Aseg = ½ R²(θ - sin θ).
How do you calculate circle radius from chord and sagitta height? +
By applying the Intersecting Chords Theorem, radius R can be derived from chord length c and sagitta height h:
R = (c² / (8h)) + (h / 2)
This is widely used in machining, surveying, and arch masonry when only the chord and depth are accessible.
What is the origin of the word "sagitta"? +
"Sagitta" is Latin for arrow. In classical geometry, the chord represents an archer's bowstring, the curved arc represents the bow, and the perpendicular height line from chord to arc resembles an arrow nocked and ready to fire.
How do you calculate circular tank liquid volume? +
When a horizontal cylindrical tank (like an oil or fuel tank) is partially full, the cross-sectional shape of the liquid is a circular segment. Multiplying the segment area by the tank's horizontal length yields the exact liquid volume in the tank.