Circular Crown Rise & Lens Sagitta

Sagitta Calculator (H = R − √(R² − (W/2)²))

Calculate the exact vertical height (sagitta H) of any circular arc from radius (R) and chord span (W), with circular segment area and central angle math.

Sagitta Rise Solver Pure Math
e.g. 30 or 36
in
e.g. 48
in
Standard Geometries: Exact Sagitta
Sagitta Height (H = Rise)
SAGITTA HEIGHT (H) Rise
12.00 in
12" (1' 0") Rise
CENTRAL ANGLE (θ) Angle
106.26 °
1.8546 radians
ARC LENGTH (L) Perimeter
55.85 in
4' 7 7/8"
SEGMENT AREA Area
402.5 sq in
2.80 sq ft
CHORD MIDPOINT TO CENTER 18.00 in (R − H)
CURVATURE RATIO (H/W) 1:4.00 (0.250)
SECTOR PIE AREA 834.5 sq in (5.80 sq ft)
Sagitta Curve Flexing Demonstrator

Interactive Sagitta Curve Flexing Simulator

Adjust the chord width to observe how the sagitta height expands as the span approaches the full circle diameter ($2R$).

48.0 Inches (H = 12.00")
Exact Sagitta Height 12.00 inches R − √(R² − W²/4)
Shallow Parabolic Approx 9.60 inches W² / 8R (Optics)
Geometric Difference +2.40 inches True circular expansion
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.

Sagitta Formulas

Sagitta Geometry & Optics Equations

Standard formulas for calculating crown rise and lens depth from radius and chord width.

1. Exact Circular Sagitta
Exact Formula: H = R − √(R² − (W/2)²)
Trig Formula: H = R × (1 − cos(θ / 2))
From Arc Length: H = R × (1 − cos(L / 2R))
Distance to Center: d = R − H = √(R² − (W/2)²)

The exact circular formula accounts for full trigonometric curvature.

2. Parabolic Approximation
Shallow Approx: H_approx ≈ W² / (8R)
Lens Sagitta (Optics): s = r − √(r² − y²)
Crown Percentage: Rise% = (H / W) × 100
Central Angle: θ = 2 × arcsin(W / 2R)

Approximation is within 1% accuracy when rise H < 0.1 × W.

Step-by-Step Worked Numeric Example

Suppose you know a circular arc has a swing radius of R = 50.0 inches and spans a chord width of W = 60.0 inches, and need the exact crown height (sagitta).

Step 1: Half-Chord (c) c = W / 2 = 60.0 / 2 c = 30.00 inches
Step 2: Apothem Distance √(50² − 30²) = √1600 d = 40.00 inches
Step 3: Sagitta Rise (H) H = R − d = 50 − 40 H = 10.00 inches
Step 4: Cross-Verification (60² + 4×10²) / 80 4000 / 80 = 50.00" (Exact Check)
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 sagitta of an arc or circle?

The sagitta (abbreviated s or H, from the Latin word for 'arrow') is the vertical distance from the center of a chord to the highest point (crown apex) of the circular arc. In optics, it represents the curve depth of a lens or mirror.

Q2 How do you calculate the sagitta from radius and chord width?

Using the Pythagorean theorem: Sagitta H = R − √(R² − (W / 2)²), where R is the radius and W is the total chord width. Alternatively, using the central angle θ: H = R × (1 − cos(θ / 2)).

Q3 How do you calculate the sagitta from arc length and radius?

First find the half-angle in radians: θ/2 = L / (2R). Then compute: Sagitta H = R × (1 − cos(L / (2R))).

Q4 What is the parabolic approximation for sagitta in optics?

For shallow curves where H << R, the sagitta can be approximated as H ≈ W² / (8R). The exact circular formula H = R − √(R² − W²/4) should always be used for architectural arches and high-precision lens manufacturing.