Trigonometric Subtended Angle Solver

Central Arch Angle Calculator (θ)

Calculate the exact central angle in degrees (°), radians (rad), and gradians (grad) from radius, arc length (L), or chord span (W).

θ = 2 arcsin(W / 2R) Trig
e.g. 30 or 36
in
e.g. 48
in
Standard Angles: Angles
Subtended Central Angle (θ)
CENTRAL ANGLE (θ) Degrees
106.26 °
1.8546 radians
ARC PERIMETER (L) R × θ
55.85 in
4' 7 7/8"
SAGITTA RISE (H) Height
12.00 in
12" (1' 0") Rise
GRADIAN ANGLE Gradians
118.07 grad
Metric angle
HALF ANGLE (θ / 2) 53.13° (0.9273 rad)
SECTOR PIE AREA 834.5 sq in (5.80 sq ft)
CURVATURE RATIO (H/W) 1:4.00 (0.250)
Interactive Protractor Dial

Interactive Radian & Degree Dial Simulator

Rotate the protractor dial to explore real-time conversion between degrees (°), radians (rad), and trigonometric sine/cosine components.

106.3° (1.855 rad)
Radian Angle (rad) 1.8546 rad θ_deg × (π / 180)
Sine Half-Angle sin(θ/2) 0.8000 W / 2R Ratio
Cosine Half-Angle cos(θ/2) 0.6000 (R − H) / R Ratio
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.

Angle Equations

Central Angle Trigonometry Formulas

Relationships between angle θ, radius R, arc length L, and chord W.

1. Angle Equations
From Arc & Radius: θ_rad = L / R
From Chord & Radius: θ = 2 × arcsin(W / 2R)
Degrees to Radians: rad = deg × (π / 180)
Radians to Degrees: deg = rad × (180 / π)

Central angle determines the circular sweep fraction of the 360° circle.

2. Associated Dimensions
Chord Length: W = 2R sin(θ / 2)
Sagitta Rise: H = R (1 − cos(θ / 2))
Sector Area: A = 0.5 × R² × θ
Segment Area: A = 0.5 × R² × (θ − sin θ)

All circular arch geometry can be uniquely solved once radius and angle are known.

Step-by-Step Worked Numeric Example

Suppose you measure a curved wall opening with a chord span of W = 48.0 inches and a vertical crown rise of H = 12.0 inches.

Step 1: Solve Radius (R) (48² + 4×12²) / 96 R = 30.00 inches
Step 2: Half-Angle Ratio sin(θ/2) = 24 / 30 sin(θ/2) = 0.8000
Step 3: Angle in Radians 2 × arcsin(0.8000) θ = 1.8546 radians
Step 4: Angle in Degrees 1.8546 × (180 / π) θ = 106.26° (106° 15')
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 What is a central angle of an arc?

A central angle (θ) is the angle subtended at the center point of a circle by the two radius arms connecting to the two endpoints of the circular arc.

Q2 How do you calculate the central angle from arc length and radius?

In radians: θ_rad = Arc Length (L) / Radius (R). In degrees: θ_deg = (L / R) × (180 / π). For example, for an arc length L = 55.85" and radius R = 30.0": θ_rad = 55.85 / 30.0 = 1.8617 radians = 106.66°.

Q3 How do you calculate the central angle from chord width and radius?

Using trigonometry: sin(θ / 2) = (Chord Width / 2) / Radius. Therefore: θ = 2 × arcsin(W / (2 × R)). For chord W = 48" and radius R = 30": sin(θ/2) = 24 / 30 = 0.80. θ/2 = 53.13°, so θ = 106.26°.

Q4 How do you convert radians to degrees and vice versa?

To convert radians to degrees: Multiply by 180 / π ≈ 57.2958. To convert degrees to radians: Multiply by π / 180 ≈ 0.0174533.