Circular Perimeter & Sector Geometry

Arc Length Calculator (L = R × θ)

Calculate the exact curved perimeter length of any circular arc from radius and angle, chord width, or sagitta height, with circular sector and segment areas.

Radius & Angle Solver Pure Math
e.g. 30 or 36
in
0° to 360°
°
Standard Circular Angles: Exact Arc
Curved Arc Perimeter (L = R × θ)
ARC LENGTH (L) R × θ
55.85 in
4' 7 7/8"
STRAIGHT CHORD (W) 2R sin(θ/2)
48.00 in
48" (4' 0") Span
SECTOR PIE AREA 0.5 R² θ
834.5 sq in
5.80 sq ft
SAGITTA RISE (H) Height
12.00 in
12" Rise Height
RADIAN MEASURE 1.8546 rad
SEGMENT AREA (ABOVE CHORD) 402.5 sq in (2.80 sq ft)
CURVATURE RATIO (H/W) 1:4.00 (0.250)
Circular Sweep Demonstrator

Interactive Central Angle Sweep Simulator

Drag the angle slider from 10° to 180° to dynamically visualize arc length expansion, chord stretch, and pie sector area scaling in real time.

106.3° Central Angle (L = 55.85")
Arc Perimeter Length 55.85 inches Curved travel length
Straight Chord Width 48.00 inches Direct endpoint span
Curved vs Straight Delta +7.85 inches (+16.4%) Perimeter elongation
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.

Trigonometric Formulas

Arc Length & Sector Equations

Standard formulas for calculating circular arcs, sectors, and chords in degrees and radians.

1. Arc Length Formulas
In Radians: L = R × θ
In Degrees: L = (π × R × θ) / 180°
Chord Length: W = 2 × R × sin(θ / 2)
Sagitta Height: H = R × (1 − cos(θ / 2))

Radians naturally equal the ratio of arc length to radius (1 rad ≈ 57.2958°).

2. Area Equations
Sector Area (Pie): A_sector = (1/2) × R² × θ
Triangle Area: A_tri = (1/2) × R² × sin(θ)
Segment Area (Cap): A_seg = (1/2) × R² × (θ − sin θ)
Full Circle Area: A_circle = π × R²

Subtracting the inner triangle area from the sector area yields the circular segment area.

Step-by-Step Worked Numeric Example

Suppose you need to determine the curved soffit perimeter for a circular arc with a swing radius of R = 36.0 inches and a central sweep angle of θ = 120°.

Step 1: Angle to Radians 120° × (π / 180) θ = 2.0944 radians (2π/3)
Step 2: Arc Length (L) L = 36.0 × 2.0944 L = 75.40" (6' 3 3/8")
Step 3: Chord Span (W) 2 × 36 × sin(60°) W = 62.35 inches
Step 4: Crown Rise (H) 36 × (1 − cos(60°)) H = 18.00 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 What is the formula to calculate circular arc length?

When angle θ is in radians: Arc Length L = R × θ. When angle θ is in degrees: Arc Length L = 2 × π × R × (θ / 360°) = (π × R × θ) / 180°. For example, for a radius R = 36" with a 90° central angle: L = (3.14159 × 36 × 90) / 180 = 56.55 inches.

Q2 How do you find arc length from chord width and height?

First calculate the radius R = (W² + 4H²) / (8H). Then find the central angle θ = 2 × arcsin(W / 2R) in radians. Finally, compute Arc Length L = R × θ.

Q3 What is the difference between an arc length and a chord length?

The chord length is the straight-line shortest distance between the two endpoints of the curve. The arc length is the actual travel distance along the curved perimeter of the circle.

Q4 How do you calculate the area of a circular sector vs. a circular segment?

A circular sector is the entire pie slice from the center (Sector Area = 0.5 × R² × θ). A circular segment is only the outer cap above the straight chord line (Segment Area = 0.5 × R² × (θ - sin θ)).