Segmental Arch Calculator
Calculate radius, sagitta, arc length, central angle, and jobsite mark-out coordinate tables for shallow circular segmental arches and pass-through openings.
Quick Definition: What is a Segmental Arch?
A segmental arch is a circular arch whose curve forms an arc of less than 180 degrees (a semicircle). Because the vertical rise is significantly less than half the horizontal span, the geometric center point of the circle falls below the springing line. Segmental arches preserve ceiling headroom in doorways and windows, but they exert substantially higher outward horizontal thrust against their supporting walls than semicircular arches, requiring robust piers or tie-rods.
Interactive Headroom Preservation Simulator
Compare walk-through clearance savings of a shallow segmental arch versus a full semicircular arch in standard 80" residential doorways.
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.
Segmental Arch Formulas & Offset Equations
Derived from the classical intersecting chord theorem where the product of the chord segments equals the product of the diameter segments: (W/2) · (W/2) = H · (2R − H).
The center drop indicates how far the center pivot point falls below the springline baseline.
Use the Y(X) equation to calculate precise vertical heights at any horizontal step distance from the center crown.
Step-by-Step Worked Numeric Example
Suppose you are building a living room cased opening with a span of W = 48 inches and a vertical crown rise of H = 12 inches.
Segmental arches maintain maximum walk-through height because the rise is shallow (often 6" to 12" on a 4-foot doorway), unlike round arches which require 24" of headroom.
Due to the 85% lateral thrust kick, always frame with double king studs and secure framing gussets to prevent wall flexing under ceiling drywall loads.
Transfer table coordinates directly onto 3/4" CDX plywood. Cut two identical curved plates to sandwich 2x4 framing blocks at 16" OC spacing.
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
Related Arch Shape Calculators
Explore complementary specialized geometry tools and layout solvers.
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
Frequently Asked Questions
Common carpentry, masonry, and geometric questions regarding arch calculations and construction layout.
Q1 What is a segmental arch?
A segmental arch is a circular arc spanning less than 180 degrees where the vertical rise is significantly less than half the span (Rise < Span / 2). The center of curvature lies below the springline baseline. It is the standard profile for interior cased pass-throughs, porch gables, and exterior brick window heads.
Q2 What is the formula to calculate the radius of a segmental arch?
Using the intersecting chords theorem: Radius R = (W² + 4H²) / (8H), where W is the horizontal span opening width and H is the crown rise (sagitta). For instance, with a 48" span and 12" rise: R = (48² + 4×12²) / (8×12) = (2304 + 576) / 96 = 30.00 inches.
Q3 Why do segmental arches exert intense horizontal thrust on side walls?
Because the arc is flatter and meets the springline at an angle rather than vertically, compressive gravitational loads resolve into strong outward diagonal vectors (often 75%–85% lateral kick). In framing, this requires double-king studs and tightly fastened structural headers.
Q4 How do you lay out a segmental arch when the center point is below the floor?
When the radius is large (e.g. 10 feet) and the center of curvature falls far below the work area, use our offset-coordinate 'Mark-Out' table. Draw a horizontal baseline, mark horizontal intervals (e.g. every 2" or 3"), measure up the calculated Y elevation at each mark, and bend a flexible batten across finish pins to trace the curve.
Q5 How do you calculate the sagitta (rise) from a given radius and span?
If you know the radius R and span W, the rise H = R - √(R² - (W/2)²). For a 60" span and a 34" radius: H = 34 - √(34² - 30²) = 34 - √(1156 - 900) = 34 - 16 = 18 inches.