True Semi-Ellipse & Focal Pin Geometry

Elliptical Arch Calculator

Calculate focal pin locations (c = √(a² − b²)), two-pin string loop lengths, Ramanujan perimeter, and shoulder vs. crown radii for wide cased openings.

Two-Foci Mathematical Ellipse Cased Openings

Total Span / Width (2a) e.g. 60 or 72

Crown Rise / Height (b) e.g. 15 or 18

Casing Moulding Width

Living Room Cased Presets: Wide Spans

True Elliptical Arch (2 Foci)

FOCAL PIN DISTANCE c = √(a²-b²)

24.00 in

24" each side of center

STRING LOOP LENGTH 2a + 2c

108.00 in

9' 0" tied nylon loop

RAMANUJAN PERIMETER L_intrados

66.86 in

5' 6 7/8" moulding

CROWN VS SHOULDER R Variable R

50.0" / 10.8"

Crown / Shoulder

FOCAL PIN SPREAD (2c) 48.00 in between pins

SEMI-ELLIPSE AREA 848.2 sq in (5.89 sq ft)

ECCENTRICITY (e) 0.800 (Oblong Oval)

Interactive Woodworking Demonstrator

Interactive String & Two-Pin Layout Simulator

Drag the angle slider to watch the taut string sweep smoothly across the two focal pins, demonstrating how constant string tension generates a true mathematical ellipse.

Tracing Pencil Position Along Curve:

Pencil Position: 90° (Crown Apex)

Pin 1 to Pencil Distance (d₁) 30.00 inches Left focal string leg

Pin 2 to Pencil Distance (d₂) 30.00 inches Right focal string leg

Constant Sum (d₁ + d₂) 60.00 in = Span Invariable sum property

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"

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

Conic Section Equations

Elliptical Arch Math & Ramanujan Formulas

The ellipse is the locus of all points where the sum of the distances to two fixed focal points ($F_1, F_2$) is a constant equal to the major axis ($2a$).

1. Focal Distance & String Length

Semi-Major Axis: a = Span / 2

Semi-Minor Axis: b = Rise (H)

Focal Pin Distance: c = √(a² − b²)

Two-Pin Loop Length: Loop = 2a + 2c

Position two finish nails at distance $c$ on either side of the centerline along the baseline.

2. Curvature Radii & Perimeter

Crown Apex Radius: R_crown = a² / b

Shoulder Radius: R_shoulder = b² / a

Ramanujan Perimeter: L ≈ π[3(a+b) − √((3a+b)(a+3b))] / 2

Semi-Ellipse Area: A = (π × a × b) / 2

Crown radius determines the flat top curvature; shoulder radius defines the tight springline curve.

Step-by-Step Worked Numeric Example

Suppose you are building a wide cased living room opening with a span of W = 60.0 inches (a = 30.0") and a crown rise of H = 18.0 inches (b = 18.0").

Step 1: Focal Distance c = √(30² − 18²) = √576 c = 24.00" from center

Step 2: String Loop Loop = 2(30) + 2(24) Loop = 108.00 inches

Step 3: Crown Radius R_top = 30² / 18 = 50.00" Flatter top curve

Step 4: Shoulder Radius R_side = 18² / 30 = 10.80" Tighter side curve

Standard Architectural Cheat Sheet

Standard Arch Opening Sizes: Quick Reference Table

Pre-calculated dimensions, radius curves, arc lengths, and material recommendations for standard residential and commercial architectural openings.

← Swipe table horizontally →

Opening Purpose	Span (W)	Rise (H) [Seg]	Radius (R)	Arc Length (L)	Center Drop	Min Casing Trim	Recommended Substrate
Interior Single Door	30.0" (2' 6")	6.0"	21.75"	33.25"	15.75"	3' 6"	1/4" High-Flex Drywall or Flex Poly
Bedroom / Bath Door	32.0" (2' 8")	8.0"	20.00"	37.10"	12.00"	3' 10"	1/4" High-Flex Drywall / Polyurethane
Main Entry / Passage	36.0" (3' 0")	9.0"	22.50"	41.74"	13.50"	4' 3"	1/4" High-Flex / Kerfed MDF Jamb
Hallway / Corridor	42.0" (3' 6")	10.5"	26.25"	48.70"	15.75"	4' 10"	Double 1/4" High-Flex Gypsum
Double Doorway / Pass-Through	48.0" (4' 0")	12.0"	30.00"	55.65"	18.00"	5' 6"	Double 1/4" High-Flex or 3/8" Gypsum
French Door Opening	60.0" (5' 0")	15.0"	37.50"	69.56"	22.50"	6' 8"	1/4" Bendable Plywood / Wiggle Board
Living Room Cased Archway	72.0" (6' 0")	18.0"	45.00"	83.47"	27.00"	8' 0"	Standard 1/2" Drywall (Wetted) or Plywood
Grand Room Divider / Great Hall	96.0" (8' 0")	24.0"	60.00"	111.30"	36.00"	10' 6"	Standard 1/2" Drywall (Dry Bend OK)

ℹ Note: All segmental dimensions calculated at standard 1:4 rise-to-span ratio ($H = W/4$). For custom ratios, use the interactive calculator above.

IRC R305 Headroom Compliant

Jobsite Carpentry & Masonry Workflow

How to Lay Out & Build an Arch: Complete Field Guide

Follow this proven 5-stage jobsite workflow to translate calculated geometry into accurate physical templates, framing, and finish trim.

Establish the Springline Datum & Rough Opening

Snap a perfectly level chalk line at the springline elevation across both jambs. Measure the total rough opening width ($W$) and verify both king studs are plumb. Mark the exact center point on the horizontal springline.

✓ Set the elevation baseline and check opening squareness.

Plot the Curve onto 3/4" Plywood Template

For circular arcs with center points within reach, swing a trammel bar from the pivot point. For large-radius or elliptical arches, mark our calculated Offset Coordinates ($X, Y$) at 2"–3" intervals, tap finish brads at each point, and bend a flexible batten across the pins to scribe the smooth line.

✓ Transfer calculated geometry to physical material.

Cut & Duplicate Arch Rib Gussets

Cut the primary arch rib out of 3/4" CDX plywood or OSB using a jigsaw. Sand the curved edge smooth to eliminate high spots. Use this master rib with a flush-trim bearing router bit to produce identical duplicate ribs for the front and back wall faces.

✓ Create rigid curved framing ribs for the soffit.

Install 2x4 Infill Blocking & Mount Rib Assembly

Fasten 2x4 blocking (cut to wall depth minus two plywood thicknesses, typically 2.0" for 2x4 walls or 5.0" for 2x6 walls) between the dual plywood ribs at 6" to 8" on center. Anchor the completed arch assembly securely into the rough opening header and king studs.

✓ Build the structural framing sandwich.

Apply Finish Substrate (Drywall, Masonry, or Flex Trim)

For drywall, fasten two layers of 1/4" High-Flex drywall to the curved soffit (dampening the backing paper with a spray bottle for tight radii). Apply flexible archway corner bead and finish mud. For trim, install pre-curved polyurethane moulding matching the calculated intrados arc length.

✓ Install the curved soffit liner and finish casing.

Essential Jobsite Tools Checklist

Required for 1/16" Precision

▸ Trammel Bar / Beam Compass

For swinging long-radius arcs onto plywood sheets with pinpoint accuracy

▸ Flexible Batten / Fairing Strip

Thin strip of PVC, vinyl, or straight-grain pine to bend across offset coordinates

▸ Framing Square & Level

To ensure a true horizontal springline baseline and plumb king studs

▸ Plumb Laser or Laser Level

For projecting centerlines and verifying vertical alignment across wide spans

▸ Jigsaw / Flush-Trim Router

To cut the master curved plywood rib template and duplicate identical gussets

▸ Bar Clamps & Finish Brads

To secure flexible drywall or trim casing tightly along the curved soffit rib

Material Engineering Matrix

Material Bending Limits & Substrate Specifications

Understand the physical minimum bend radii and structural limitations of gypsum boards, bendable plywood, kerfed jambs, and flexible trim.

Drywall & Plaster ● Recommended for Tight Arches

1/4" High-Flex Gypsum Board

Min Dry Radius 24" (610 mm)

Min Wet Radius 12" (305 mm)

Specially engineered core with high tensile facing paper. Apply in two laminated 1/4" layers for 1/2" total thickness.

Fastener: 1-1/4" Type W/S Screws at 6" O.C.

Drywall & Plaster ● Standard Residential Spans

3/8" Standard Drywall

Min Dry Radius 60" (1,525 mm)

Min Wet Radius 36" (915 mm)

Moderate flexibility. Suitable for wide residential pass-throughs and hallway arches with radius over 5 feet.

Fastener: 1-1/4" Coarse Thread Screws

Drywall & Plaster ● Gentle Arches Only

1/2" Standard Drywall

Min Dry Radius 120" (3,050 mm)

Min Wet Radius 72" (1,830 mm)

Rigid board. Must not be bent dry under 10 ft radius. For wet bending, spray both sides with water and allow to soak for 15 minutes.

Fastener: 1-5/8" Drywall Screws

Wood & Millwork ● Best for Wood Arch Jambs

1/4" Bendable Plywood (Wiggle Wood)

Min Dry Radius 8" to 12" (200–305 mm)

Min Wet Radius N/A (Dry Only)

Multi-ply rotary cut veneer with cross-grain plies omitted. Available in Column Bend (8-ft axis) and Barrel Bend (4-ft axis).

Fastener: 18-Gauge Brads + Construction Adhesive

Wood & Millwork ● Custom Cased Openings

Kerfed 3/4" MDF or Clear Pine

Min Dry Radius 14" (355 mm)

Min Wet Radius N/A (Dry Only)

Saw kerfs cut to 5/8" depth (leaving 1/8" face veneer) spaced 1/2" to 3/4" apart. Fill kerfs with PVA glue during mounting for rigidity.

Fastener: Finish Screws into 2x4 Rib Blocking

Finish Trim & Casing ● Standard Finish Casing

Flexible Polyurethane Moulding

Min Dry Radius 12" to 18" (305–455 mm)

Min Wet Radius N/A (Dry Only)

Seamless imitation of wood grain that flexes effortlessly along the intrados. Order based on calculated Arc Length + 10%.

Fastener: 15-Gauge Finish Nails + Polyurethane Adhesive

Structural Mechanics & Physics

Structural Thrust Dynamics & Lateral Force Engineering

Understand how gravitational loads resolve into outward lateral forces, and how to size king studs, headers, and abutments to prevent framing deflection.

Thrust Equation

The Fundamental Law of Arch Thrust

The horizontal outward thrust (H_thrust) generated at each springline abutment is inversely proportional to the rise height (H):

H_thrust = (Total Load × Span) / (8 × Rise)

Key Rule: Halving the rise height doubles the outward lateral force pushing against your king studs or masonry abutment piers!

Framing Reinforcement Rules

✓ Span ≤ 48": Double 2x4 king studs each side

✓ Span 48"–72": Triple king studs + 2x8 structural header

✓ Span > 72": Engineered LVL header with structural tie straps

✓ Masonry Arches: Skewback width must be ≥ 1.5× wall thickness

Low Lateral Thrust

Semicircular (Roman) Arch

Lateral Thrust Ratio 20% – 30%

Because the curve meets the springline at a true vertical 90° tangent, the majority of dead load is directed vertically down into the supporting jambs.

Vector: Primarily Vertical (Downwards)

Moderate Lateral Thrust

Gothic (Pointed) Arch

Lateral Thrust Ratio 35% – 50%

The pointed apex divides the curve into steeper arcs, reducing horizontal outward vector forces compared to flatter round arches.

Vector: Steep Diagonal Vector

High Lateral Thrust

Segmental (Shallow) Arch

Lateral Thrust Ratio 75% – 90%

As the rise decreases relative to the span, the thrust vector angles sharply outward, exerting immense lateral force on the supporting side walls.

Vector: Strong Outward Diagonal Kick

Severe Lateral Thrust

Jack (Flat) Lintel Arch

Lateral Thrust Ratio 150% – 200%

Flat arches rely solely on wedge friction and severe horizontal thrust to hold the voussoirs aloft. Requires massive abutments or structural tie rods.

Common Mistake:

Measuring the arch rise from the finished floor without verifying that the left and right springline points are level, resulting in a tilted, lopsided arch crown.

Pro Trade Solution:

Shoot a laser level line across the opening to establish an absolute level springline datum before tracing or mounting curved plywood ribs.

Field Master Pro-Tips for Clean Execution

★ Pre-curve flexible drywall overnight around a cylindrical form before hanging to relax internal surface tension.

★ When making plywood rib templates, clamp both ribs together and sand simultaneously for identical profiles.

★ Always leave centering formwork and shoring wedges in place until masonry mortar achieves full 7-day compressive cure strength.

★ For flexible casing moulding, acclimatize the material to room temperature (70°F+) for 24 hours prior to bending.

Complete Tool Index

Arch Calculators & Trade Solvers Directory

Quickly filter across our specialized trade tools, pure geometry solvers, and architectural reference guides.

Arch Shape Calculators

Tier 1 Shapes (9)

Trade & Jobsite Tools

Tier 2 Trades (10)

Geometric Solvers & Math

Tier 3 Math (5)

Authority References

Tier 4 Guides (4)

Expert Knowledge Base

Frequently Asked Questions

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

Q1 How do you calculate the focal pin distance for drawing an elliptical arch?

The focal distance from the center point to each pin is c = √(a² - b²), where a is the semi-major axis (half the span, W/2) and b is the semi-minor axis (the crown rise H). For example, for an arch with span W = 60" (a = 30") and rise H = 18" (b = 18"): c = √(30² - 18²) = √(900 - 324) = √576 = 24.00 inches from center on each side.

Q2 What is the total string loop length needed for the two-pin method?

For a tied closed loop enclosing both pins and the tracing pencil, the total loop length is Loop = 2a + 2c. Alternatively, if tying the string to the two pins, the cut string length between the pins is String Length = 2a (the full span width).

Q3 What is the difference between an elliptical arch and a segmental arch?

A segmental arch has a single, constant radius of curvature across its entire curve. An elliptical arch has variable curvature—it curves sharply at the springline shoulders (radius R = b²/a) and flattens out smoothly at the crown apex (radius R = a²/b), creating a wide, elegant opening with minimal vertical rise.

Q4 How accurate is Ramanujan's formula for elliptical arch perimeter?

Ramanujan's second approximation formula for ellipse perimeter has an error of less than 0.0001% for all residential arch proportions, providing exact linear lengths for ordering flexible casing moulding and drywall liner strips.