Three-Centered Arch Calculator
Calculate shoulder radius (R₁), crown radius (R₂), center coordinates, and smooth tangent transition points for basket-handle openings, wide masonry portals, and interior cased passageways.
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.
| Pt # | Dist from Center (X) | Height (Y) | Fraction | Drop |
|---|
Live Plywood Scribing Simulator
🔨 Carpentry Technique: Tap finish brads at coordinate marks. Bend a flexible batten against the pins to scribe a smooth curve.
Standard Arch Opening Sizes: Quick Reference Table
Pre-calculated dimensions, radius curves, arc lengths, and material recommendations for standard residential and commercial architectural openings.
| Opening Purpose | Span (W) | Rise (H) | Radius (R) | Arc Length (L) | Center Drop | Min Casing Trim | Recommended Substrate |
|---|---|---|---|---|---|---|---|
| Interior Single Door | 30.0" | 6.0" | 21.75" | 33.25" | 15.75" | 3 ft 6 in | 1/4" High-Flex Drywall or Flex Poly |
| Bedroom / Bath Door | 32.0" | 8.0" | 20.00" | 37.10" | 12.00" | 3 ft 10 in | 1/4" High-Flex Drywall / Polyurethane |
| Main Entry / Passage | 36.0" | 9.0" | 22.50" | 41.74" | 13.50" | 4 ft 3 in | 1/4" High-Flex / Kerfed MDF Jamb |
| Hallway / Corridor | 42.0" | 10.5" | 26.25" | 48.70" | 15.75" | 4 ft 10 in | Double 1/4" High-Flex Gypsum |
| Double Doorway / Pass-Through | 48.0" | 12.0" | 30.00" | 55.65" | 18.00" | 5 ft 6 in | Double 1/4" High-Flex or 3/8" Gypsum |
| French Door Opening | 60.0" | 15.0" | 37.50" | 69.56" | 22.50" | 6 ft 8 in | 1/4" Bendable Plywood / Wiggle Board |
| Living Room Cased Archway | 72.0" | 18.0" | 45.00" | 83.47" | 27.00" | 8 ft 0 in | Standard 1/2" Drywall (Wetted) or Plywood |
| Grand Room Divider / Great Hall | 96.0" | 24.0" | 60.00" | 111.30" | 36.00" | 10 ft 6 in | Standard 1/2" Drywall (Dry Bend OK) |
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.
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.
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.
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.
Apply Finish Substrate (Drywall, Flex Trim, or Casing)
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.
Material Bending Limits & Substrate Specifications
Understand the physical minimum bend radii and structural limitations of gypsum boards, bendable plywood, kerfed jambs, and flexible trim.
1/4" High-Flex Gypsum Board
Specially engineered core with high tensile facing paper. Apply in two laminated 1/4" layers for 1/2" total thickness.
3/8" Standard Drywall
Moderate flexibility. Suitable for wide residential pass-throughs and hallway arches with radius over 5 feet.
1/2" Standard Drywall
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.
1/4" Bendable Plywood (Wiggle Wood)
Multi-ply rotary cut veneer with cross-grain plies omitted. Available in Column Bend (8-ft axis) and Barrel Bend (4-ft axis).
Kerfed 3/4" MDF or Clear Pine
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.
Flexible Polyurethane Moulding
Seamless imitation of wood grain that flexes effortlessly along the intrados. Order based on calculated Arc Length + 10%.
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.
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):
Key Rule: Halving the rise height doubles the outward lateral force pushing against your king studs or masonry abutment piers!
General residential partition guidelines. For load-bearing walls, second-story floor loads, or heavy masonry, consult a structural engineer and local building codes (IRC R602).
Semicircular (Roman) Arch
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.
Gothic (Pointed) Arch
The pointed apex divides the curve into steeper arcs, reducing horizontal outward vector forces compared to flatter round arches.
Segmental (Shallow) Arch
As the rise decreases relative to the span, the thrust vector angles sharply outward, exerting immense lateral force on the supporting side walls.
Jack (Flat) Lintel Arch
Flat arches rely solely on wedge friction and severe horizontal thrust to hold the voussoirs aloft. Requires massive abutments or structural tie rods.
Common Arch Doorway Framing & Casing Mistakes
Prevent unlevel springlines, irregular cripple stud lengths, and misaligned trim casing.
Exceeding Drywall Minimum Bending Radius
Attempting to force standard 1/2" drywall around tight curves (under 10 ft radius) dry, causing catastrophic gypsum core fracture and unsightly faceted creases.
Use two layers of 1/4" High-Flex gypsum (bends dry to 24" radius, wet to 12" radius). For ultra-tight arches, use kerfed 3/4" MDF or 1/4" bendable plywood (wiggle board).
Underestimating Lateral Horizontal Thrust
Treating shallow segmental or flat jack arches like standard post-and-beam lintels without reinforcing the side supporting piers or king studs.
Install double or triple king studs tightly fastened with structural framing screws. For masonry arches, ensure skewbacks are angled toward the radius center with stout abutments.
Ordering Casing by Span Width Instead of Arc Length
Ordering flexible polyurethane or wood moulding based on the horizontal rough opening width ($W$), leaving the installer short by 15% to 57%.
Always order moulding based on the true calculated Intrados Arc Length (L_intrados) plus a minimum 10% allowance for trimming springline plumb cuts.
Uneven Springline Heights Across Jambs
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.
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
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 (10)- Semicircular (Roman) Calculator→
Classic 180° half-circle arches with vertical springlines
- Segmental Arch Calculator→
Shallow circular chord arches for doorways & lintels
- Elliptical Arch Calculator→
True semi-ellipses and 3/5-centered compound curves
- Gothic Pointed Arch Calculator→
Equilateral and lancet two-centered pointed arches
- Tudor (Four-Centered) Calculator→
Compound 4-radius flat-pointed English Tudor arches
- Horseshoe Moorish Calculator→
Keyhole arches with sweep angles exceeding 180°
- Jack (Flat) Lintel Calculator→
Horizontal flat lintels with angled skewback joints
- Parabolic Arch Calculator→
Quadratic curves for uniform vertical deck loads
- Catenary Arch Calculator→
Hyperbolic pure-compression hanging chain curves
- Three-Centered Arch Calculator→
Basket-handle 3-radius compound curves for wide portals
Trade & Jobsite Tools
Tier 2 Trades (12)- Brick Voussoir & Keystone Tool→
Odd brick counts, skewback angles & mortar joint taper
- Arch Window & Glazing Sizing→
Rough openings for Andersen/Marvin glass & vinyl trim
- Doorway Framing & Plywood Ribs→
2x4 cripple stud drop heights & plywood gusset templates
- Trim & Flexible Moulding Solver→
Intrados vs. extrados casing lengths & springline miters
- Structural Arch & Bridge Tool→
Horizontal thrust, load vectors & abutment sizing
- Arch Materials & Cost Estimator→
Plywood sheet nesting, 2x4 lumber takeoffs & BOM
- Formwork & Centering Calculator→
Timber ribs, lagging slats & striking wedge clearances
- Drywall Arch Bending Limits→
Minimum radii for 1/4" High Flex & kerfed MDF spacing
- Garden Arbor & Trellis Calculator→
Curved pergola rafters, bent-lam strips & headroom
- Pipe-Arch Culvert Calculator→
Corrugated steel pipe-arch area & Manning flow capacity
- Pizza Oven Arch Calculator→
Firebrick counts & 63% door-to-dome thermodynamic ratio
- Barrel Vault Ceiling Calculator→
Curved ceiling framing ribs at 16" O.C. & drywall takeoffs
Geometric Solvers & Math
Tier 3 Math (5)- Arc Radius (from Width & Height)→
Solve radius R from chord width and sagitta height
- Arc Length & Perimeter Solver→
Calculate curved perimeter distance from radius and angle
- Arch Central Angle Calculator→
Compute subtended sweep angle in degrees and radians
- Sagitta (Arch Rise) Calculator→
Calculate vertical rise height from span and radius
- Complete Arc Solver (Any 2 Knowns)→
Enter any 2 parameters to solve all 5 dimensions
Authority References
Tier 4 Guides (6)- Arch Terminology & Anatomy Glossary→
Illustrated guide to keystone, voussoir, extrados & impost
- Types of Arches Full Comparison→
Comprehensive structural, historical and visual matrix
- How to Draw Arches on Plywood→
String compass, beam trammel & stepping stick methods
- Types of Arches & Selection Guide→
Decision framework for ceiling height, framing & aesthetics
- How to Build a Brick Arch→
5-stage masonry guide on voussoir layout, centering & wedges
- How to Bend Drywall Around an Arch→
Dry flexing, wet sponging & back-kerfing for tight curves
- Calculate Arch Radius from Width & Height→
Pythagorean chord theorem proof & carpenter fraction math
Frequently Asked Questions
Common carpentry, masonry, and geometric questions regarding arch calculations and construction layout.
Q1 What is a three-centered (basket-handle) arch?
A three-centered arch (also known as a basket-handle arch, oval arch, or anse de panier) is a depressed compound arch constructed from three tangent circular arcs. Two tighter shoulder arcs curve up from the springline with radius R₁, while a flatter central crown arc with radius R₂ connects them smoothly across the apex.
Q2 How does a three-centered arch differ from a true ellipse?
A true ellipse has a continuously varying radius that requires string loops or trammel points to lay out. A three-centered arch closely approximates the graceful aesthetic of an ellipse but is composed entirely of circular arcs that can be struck with a standard compass or trammel beam on jobsite plywood.
Q3 How do you calculate the crown radius (R₂) from span and rise?
Given span W, rise H, and chosen shoulder radius R₁: Crown Radius R₂ = (W² / 4 + H² − 2 × R₁ × H) / (2 × (H − R₁)). The transition point occurs where the shoulder circle and crown circle meet along the line connecting their centers.
Q4 Where are three-centered arches used in construction?
They are widely used for wide interior cased doorways, brick fireplace openings, carriage doors, and masonry vaults where maximum vertical headroom is desired across the central opening.