Universal 2-of-5 Circular Arc Solver

Complete Circular Arc Solver

Enter any 2 known values (W, H, R, θ, L) to instantly calculate all remaining parameters, sector/segment areas, and offset mark-out tables.

in
in
Standard Test Pairs: Instant Solve
2-of-5 Analytical Solution Found
1. SPAN (W)
48.00"
2. RISE (H)
12.00"
3. RADIUS (R)
30.00"
4. ANGLE (θ)
106.26°
5. ARC (L)
55.85"
CIRCULAR SEGMENT AREA 402.5 sq in (2.80 sq ft)
SECTOR PIE AREA 834.5 sq in
SOLVER METHOD Closed-Form Analytic
Field Layout Generator

Offset-Coordinate "Mark-Out" Table & Interactive Plotter

For large arches where the swing radius exceeds your compass or room length. Simply draw your baseline, measure out horizontal intervals from center (X), and mark vertical height (Y) to plot perfect curves directly on wood or wall.

Coordinate Stepping Points (Centerline Outward)
Based on current Span: 48" • Rise: 12" • Radius: 30.00"
Step Interval:
Pt # Dist from Center (X) Height (Y) Fraction Drop
💡 Tip: Symmetrical — use identical Y heights for both left and right sides.

Live Plywood Scribing Simulator

Point 1 of 13
Plotted Offset (X): 0.00" (Center Apex)
Height Y: 12.000" (12") Drop: 0.000"
3/4" Plywood Blank Baseline X-Axis (Springline) Y: 12"

🔨 Carpentry Technique: Tap finish brads at coordinate marks. Bend a flexible batten against the pins to scribe a smooth curve.

Canonical Equations

The 5 Canonical Circular Arc Relationships

All circular segments are governed by these 5 interrelated mathematical identities.

1. Core Relationships
Radius from W & H: R = (W² + 4H²) / (8H)
Chord from R & θ: W = 2R sin(θ / 2)
Rise from R & θ: H = R (1 − cos(θ / 2))
Arc Length: L = R × θ (in radians)

Knowing any 2 parameters locks the remaining 3 degrees of freedom.

2. Transcendental Numerical Solving
Target Function: f(θ) = 2 sin(θ/2) / θ − W/L = 0
Derivative: f'(θ) = [θ cos(θ/2) − 2 sin(θ/2)] / θ²
Newton Step: θ_(n+1) = θ_n − f(θ_n) / f'(θ_n)
Radius: R = L / θ

Newton-Raphson rapidly converges within 4 iterations to double-precision floating point accuracy.

Step-by-Step Worked Numeric Example

Suppose you know only two parameters for a circular segment: chord width W = 40.0 inches and radius R = 25.0 inches.

Step 1: Central Angle (θ) 2 × arcsin(20 / 25) θ = 106.26° (1.8546 rad)
Step 2: Sagitta / Rise (H) 25 − √(25² − 20²) H = 10.00 inches
Step 3: Arc Length (L) 25.0 × 1.8546 L = 46.36 inches
Step 4: Segment Area (A) (25²/2) × (1.8546 − sin θ) A = 279.56 sq inches
Trade Gotchas & Quality Control

Common Arc Geometry & Trigonometry Calculation Errors

Prevent severe dimensional errors when converting radians, solving sagittas, or computing curved arc segments.

⚠️ TRIGONOMETRY

Mixing Degrees and Radians in Arc Length Formulas

Common Mistake:

Using angle in degrees directly in $L = R imes heta$ without converting to radians, producing an arc length 57 times larger than reality.

Pro Trade Solution:

Always multiply degree angles by $pi / 180$ before computing arc length or sector dimensions, or use our direct degree formula $L = pi imes R imes heta / 180$.

⚠️ CHORD THEOREM

Assuming Crown Rise Equals Circle Radius

Common Mistake:

Assuming rise ($H$) equals radius ($R$) for shallow segmental arches. This is only true for a full 180° semicircular arch ($H = W/2$).

Pro Trade Solution:

Always calculate radius using the intersecting chord formula $R = (W^2 + 4H^2) / (8H)$ whenever rise is less than half the horizontal opening width.

⚠️ PRECISION ROUNDING

Premature Decimal Rounding on Coordinate Offsets

Common Mistake:

Rounding intermediate radius or angle values to 1 decimal place, resulting in cumulative layout errors exceeding 1/2" at the arch shoulders.

Pro Trade Solution:

Carry calculations to at least 4 decimal places internally before converting final jobsite mark-out measurements to 1/16" carpenter fractions.

⚠️ CHORD VS ARC

Ordering Materials by Chord Width Instead of Arc Length

Common Mistake:

Measuring straight horizontal chord span ($W$) to order flexible jamb liners or trim, leaving the installer significantly short on material.

Pro Trade Solution:

Calculate true Intrados Arc Length ($L$) along the curved soffit perimeter and add 10% for end trimming and miter joints.

Field Master Pro-Tips for Clean Execution

★ Verify subtended angles in both radians and degrees to avoid calculation errors in downstream CAD or CNC software.
★ For compound arches, compute focal distance offsets using double-precision floating point math before rounding to jobsite fractions.
★ Check that chord width does not exceed twice the radius (W <= 2R), as real geometric solutions do not exist beyond the circle diameter.
★ Cross-check calculated arc length against chord width; arc length must strictly exceed horizontal span width for all curved profiles.
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Expert Knowledge Base

Frequently Asked Questions

Common carpentry, masonry, and geometric questions regarding arch calculations and construction layout.

Q1 How does the 2-of-5 Arc Solver work?

A circular arc is uniquely defined by exactly 2 independent geometric variables (such as Radius and Span, Span and Height, or Arc Length and Radius). By selecting any 2 known parameters, this calculator uses exact algebraic equations or Newton-Raphson numerical methods to solve all remaining dimensions.

Q2 What are the 10 possible pairs of known parameters?

The 10 possible 2-variable combinations are: (1) Span & Height [W, H]; (2) Radius & Span [R, W]; (3) Radius & Height [R, H]; (4) Radius & Angle [R, θ]; (5) Radius & Arc Length [R, L]; (6) Span & Angle [W, θ]; (7) Height & Angle [H, θ]; (8) Arc Length & Angle [L, θ]; (9) Span & Arc Length [W, L]; (10) Height & Arc Length [H, L].

Q3 Why do Span & Arc Length (W, L) require numerical solving?

The relationship between Chord W and Arc Length L with unknown radius R leads to the transcendental equation sin(L / 2R) = W / 2R, which cannot be isolated with elementary algebra. We use Newton-Raphson iteration to solve for radius R to 10 decimal places of accuracy in under 1 millisecond.

Q4 Is there any combination of 2 parameters that cannot be solved?

All 10 combinations of the 5 canonical circular arc variables have unique geometric solutions, provided the inputs satisfy basic physical constraints (e.g. Arc Length L ≥ Chord Width W, and Height H ≤ Radius R).