02 // Visual Math & Finance Solverss = Rθ, c = 2R·sin(θ/2), A_sector = ½R²θ, A_seg = ½R²(θ − sin θ)
Visual Circle Sector, Arc Length, Chord & Sagitta Geometry Solver
A circular sector of radius $R$ and central angle $\theta$ (in radians) subtends arc length $s = R\theta$, chord length $c = 2R\sin(\theta/2)$, sector area $A_{\text{sector}} = \frac{1}{2}R^2\theta$, and circular segment area $A_{\text{seg}} = \frac{1}{2}R^2(\theta - \sin\theta)$.
s = Rθ, c = 2R·sin(θ/2), A_sector = ½R²θ, A_seg = ½R²(θ − sin θ)Inputs & Parameters (2)
m
1m±0.550m
°
5°±1355°
Sector Area (½R²θ)
138.23 m²
30.6% of full circle (452.4 m²)
Arc Length (s = Rθ)
23.04 m
Full circumference = 75.4 m
Chord Length (c)
19.66 m
Sagitta height h = 5.12 m
Circular Segment Area
70.57 m²
Apothem d = 6.88 m
20-Row Comparison Table
Compare across:
| # | Circle Radius (R) (m) | Sector Area | Arc Length s | Chord Length c |
|---|---|---|---|---|
| 01 | 1 m | 0.96 m² | 1.92 m | 1.64 m |
| 02 | 3.5 m | 11.759 m² | 6.72 m | 5.73 m |
| 03 | 6 m | 34.558 m² | 11.52 m | 9.83 m |
| 04 | 8.5 m | 69.355 m² | 16.32 m | 13.93 m |
| 05 | 11.5 m | 126.951 m² | 22.08 m | 18.84 m |
| 06 | 14 m | 188.146 m² | 26.88 m | 22.94 m |
| 07 | 16.5 m | 261.341 m² | 31.68 m | 27.03 m |
| 08 | 19 m | 346.535 m² | 36.48 m | 31.13 m |
| 09 | 21.5 m | 443.728 m² | 41.28 m | 35.22 m |
| 10 | 24 m | 552.92 m² | 46.08 m | 39.32 m |
| 11 | 27 m | 699.79 m² | 51.84 m | 44.23 m |
| 12 | 29.5 m | 835.38 m² | 56.64 m | 48.33 m |
| 13 | 32 m | 982.969 m² | 61.44 m | 52.43 m |
| 14 | 34.5 m | 1,142.558 m² | 66.24 m | 56.52 m |
| 15 | 37 m | 1,314.146 m² | 71.03 m | 60.62 m |
| 16 | 39.5 m | 1,497.732 m² | 75.83 m | 64.71 m |
| 17 | 42.5 m | 1,733.876 m² | 81.59 m | 69.63 m |
| 18 | 45 m | 1,943.86 m² | 86.39 m | 73.72 m |
| 19 | 47.5 m | 2,165.845 m² | 91.19 m | 77.82 m |
| 20 | 50 m | 2,399.828 m² | 95.99 m | 81.92 m |
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Key Analytical Takeaways
- 01The chord $c = 2R\sin(\theta/2)$ connects the two arc endpoints in a straight line, splitting the pie sector into an isosceles triangle plus a shaded circular segment.
- 02The apothem $d = R\cos(\theta/2)$ measures distance from the center to the chord midpoint, while the sagitta (versine height) $h = R - d$ measures the arch rise.
- 03Civil engineers and woodworkers can reconstruct radius $R$ from chord $c$ and sagitta $h$ using $R = \frac{c^2}{8h} + \frac{h}{2}$.
Frequently Asked Questions
What is the difference between a circle sector and a circle segment?+
A sector is the entire pie slice bounded by two radii and the arc. A circular segment is only the cap region between the straight chord and the outer arc (Sector Area minus Triangle Area).
How do you find the radius of an arch if you only know its width (chord c) and height (sagitta h)?+
By the intersecting chords theorem, R = c² / (8h) + h / 2.
What is the sagitta (versine height) of a circular arc?+
The sagitta h = R(1 − cos(θ/2)) is the perpendicular rise from the midpoint of the chord up to the outer arc crown—crucial in bridge arches, optical lenses, and CNC machining.