Arc-to-chord divergence and area growth across central angleLive current inputs
Math & Statistics
Circle Value Table Calculator
Generate an angle-by-angle circle geometry table for one radius, compare arc and chord growth, and locate the selected central angle on live response curves for sector, segment, and sagitta.
Generated angle rows-
Selected arc length-
Selected chord length-
Selected sector area-
Selected segment area-
Selected sagitta-
Arc minus chord-
Segment share of sector-
ANGLE RESPONSE ATLAS
Arc-to-chord divergence and area growth across central angle
Two normalized response panels preserve each quantity's unit while a selected-angle cursor links the chart to the exact highlighted table row.
LIVE VALUE TABLE
Circle quantities at every requested angle increment
The selected angle is inserted even when it does not land exactly on the regular increment.
| Angle | Arc length | Chord | Sagitta | Sector area | Segment area | Arc - chord |
|---|
TABLE DESIGN
Choose a range and increment that answer the comparison question
- Enter one positive radius.
- Set start and end angles in increasing order.
- Choose an increment small enough to show the response shape.
- Enter a selected angle for exact chart and table emphasis.
- Compare quantities only with compatible units or normalized scales.
WHY THE CURVES SEPARATE
Arc grows linearly while chord growth eventually flattens
Arc length is directly proportional to angle in radians. Chord length depends on sine and approaches the diameter at 180 degrees, so the arc-minus-chord gap grows.
Segment area is the sector area minus the center triangle. At small angles it is a small fraction of the sector, then its share grows as curvature becomes more pronounced.
PARAMETRIC ANGLE SWEEP
Hold radius fixed and evaluate every angle independently
Each degree value is converted to radians and passed through the exact circle formulas. The end point and selected point are retained even when the increment does not divide the interval.
Detailed calculation process and general formulas
θᵣ=θπ/180L(θ)=rθᵣq(θ)=2r sin(θᵣ/2)g(θ)=r[1-cos(θᵣ/2)]Kseg(θ)=r²(θᵣ-sin θᵣ)/2Symbols, meanings, and units
- θ
- one table central angledegrees
- θᵣ
- central angle in radiansradians
- L
- arc lengthlength
- q
- chord lengthlength
- g
- sagittalength
- Kseg
- circular-segment arealength²
TABLE READING
Three comparisons reveal different geometric effects
The selected row anchors the live curves to exact quantities.
Boundary divergence
-Arc minus chord quantifies the cost of following the curve instead of spanning endpoints.
Cap depth
-Sagitta tracks how far the arc bows away from the chord.
Area composition
-Segment share shows how much of the sector lies beyond the chord.
Decision takeaway: Use the table for exact takeoff values and the curves for sensitivity to angle changes.
Applied decisions
Two angle sweeps with practical meaning
Arch family study
A designer compares several included angles for one fabrication radius.
What the result clarifies: Chord, rise, and curved-edge length can be reviewed in the same row.
Sector coverage alternatives
A rotating device can operate through several sweep angles.
What the result clarifies: Sector and segment growth show how coverage changes nonlinearly.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
The table uses ideal circle geometry and does not include thickness, kerf, overlap, chord offsets, or manufacturing constraints. Extremely small increments are capped to a practical row count for browser performance.
Circle Value Table Calculator | Arc, Chord, Sector, Segment, and Sagitta by Angle FAQ
Why is the selected angle added to the table?
It preserves an exact decision row even when the regular increment would skip it.
Why normalize some chart curves?
Lengths and areas have different dimensions; normalization allows shape comparison without pretending their raw values are comparable.
Can the range exceed 360 degrees?
No. The page models one central angle on a single circle.
Why does chord length flatten?
Chord length follows 2r sin(θ/2), reaching a diameter at 180 degrees.