Arch Calculator
Instantly compute radius, swing chord, central angle, voussoir brick counts, and field-ready offset-coordinate mark-out tables. Built for carpenters, masons, and architects.
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.
What Is an Arch Calculator?
An Arch Calculator is a specialized construction geometry tool designed to solve the physical dimensions required to draw, frame, template, or build curved architectural structures from span and rise.
In construction trades, manual calculations often introduce compounding errors. A difference of just 1/4" in sagitta can cause flexible moulding to buckle or masonry voussoirs to misalign.
Interactive Chord & Radius Simulator
💡 Field Insight: Center drops below springline baseline as rise gets shallower.
How to Use the Arch Calculator
From taking rough opening measurements on the jobsite to stepping out coordinate curves onto 3/4" plywood templates in 4 straightforward steps.
Profile & Trade Configuration
Click different arch profiles below to preview how our geometry engine switches calculation formulas and trade layout rules:
How Arch Calculator Works — Formulas & Equations
Clean geometric equations with real-time interactive vector annotations. Click or hover any formula variable to highlight its physical dimension.
Euclidean Chord Theorem Geometry
The radius of the circular arc. Because the center of curvature lies below the springline baseline, the swing arm extends distance R from pivot (C) to apex (A).
Radius from Span & Rise
Derived from Euclid's intersecting chords theorem:
Y(X) = √(R² - X²) - (R - H)
Hover over variables above to highlight corresponding geometry.
Arc Length & Central Angle
Calculate the perimeter curve distance along the soffit and total angular sweep:
θ (deg) = θ (rad) × (180 / π)
L (Arc) = R × θ (rad)
Essential for ordering flexible trim, drywall bead, or bending formwork ribs.
Elliptical Axes & Focal Points
Semi-major axis a, semi-minor axis b, and pin focus distance c:
c (Focus) = √(a² - b²)
P ≈ π × [3(a+b) - √((3a+b)(a+3b))]
String length for layout equals full span (W).
Dynamic Worked Example: Test Your Custom Values
Apply chord theorem:
R = 45.00 inches
Half-chord arcsine:
θ = 106.26° (1.855 rad)
Radius × Radians:
L ≈ 83 7/16" (83.48")
Cut dimensions:
Radius Center: -27.0" drop
7 Essential Architectural Arch Types & Thrust Simulator
Compare mathematical formulas, structural thrust characteristics, and live compressive load vectors across the 7 fundamental arch profiles.
Segmental Arch Profile
Shallow circular chord arches create intense horizontal outward forces (thrust) at the springline, requiring substantial wall backing or reinforced abutments.
Segmental Arch
Shallow circular arc less than 180°
Semicircular (Roman) Arch
Full 180° half-circle where Rise = Half Span
Elliptical Arch
True semi-ellipse or 3/5-centered compound curve
Gothic (Pointed / Equilateral)
Two intersecting circular arcs meeting at an apex point
Tudor (Four-Centered) Arch
Compound 4-radius curve with pointed apex and low rise
Horseshoe (Moorish / Keyhole)
Circular arc exceeding 180° with inwards-curving imposts
Jack / Flat Lintel Arch
Horizontal soffit supported by angled skewback voussoirs
Types of Arches Comparison Matrix & Live Inspector
Compare geometric construction, structural load characteristics, and framing difficulty across all architectural arch shapes.
| Arch Style | Geometry Base | Rise-to-Span | Structural Thrust | Difficulty | Typical Application |
|---|---|---|---|---|---|
| ◒ Segmental | Single shallow circular arc (center below springline) | 1:3 to 1:6 | High lateral kick (requires abutments) | Easy | Interior room dividers, brick window heads |
| ◐ Semicircular (Roman) | Full 180° semicircle (Radius = Span/2) | Exact 1:2 | Balanced vertical load (zero kick) | Easy | Classical architecture, Roman aqueducts, arches |
| ⬭ Elliptical | True mathematical ellipse (major & minor axes) | Flexible | Smooth compression distribution | Moderate | Cased openings, luxury custom trim, carriage doors |
| ⋀ Gothic / Pointed | Dual intersecting circular arcs meeting at crown | 1:1 to 2:1 | Directs thrust sharply downward | Moderate | Cathedrals, ecclesiastical windows, tall gates |
| ⌒ Tudor (4-Centered) | Four distinct radius centers (2 tight, 2 flat) | 1:3 | Moderate lateral thrust | Advanced | English Tudor fireplaces, estate entry gates |
| ☉ Horseshoe (Moorish) | Circular arc extending beyond 180° | > 1:2 | Inward pinching at neck springline | Advanced | Islamic, Moorish, and Spanish revival architecture |
| ▭ Jack / Flat Arch | Flat horizontal soffit with angled skewback bricks | 0 (Flat) | Extremely high lateral kick (Requires Buttress) | Advanced | Georgian brick window lintels, fireplace surrounds |
Best for: Interior room dividers, brick window heads, shallow porch headers.
Anatomy of an Arch — Terminology & Geometry
Click or hover over any term or interact with diagram hotspots to inspect key architectural components.
The central wedge-shaped stone or framing gusset at the crown locking the arch into compression.
The central wedge-shaped stone or framing gusset at the crown locking the arch into compression.
The inner, bottom concave surface of the arch opening determining clear walk-through height.
The upper, exterior convex curved boundary of the arch ring or framing plate.
The wedge-shaped masonry units or segmented framing blocks forming the curved arch ring.
The imaginary horizontal baseline from which the curve begins springing away from vertical jambs.
The inclined or horizontal seating surface on piers bearing outward diagonal arch thrust.
Material Limits & Jobsite Construction Factors
Designing an arch is only half the battle. Building it requires accounting for physical material bend limits, formwork deflection, and masonry thrust lines.
Drywall & Trim Bending Stress Simulator
1/4" High-Flex Drywall can bend completely dry to a 30" radius without wetting or steaming.
Minimum Bending Radii for Soffit Linings
Gypsum Assoc. GA-216Never attempt to bend sheet materials beyond manufacturer elastic limits. If your calculated arch radius is tighter than the minimums below, kerf the substrate or use multi-layered flexible gypsum:
| Material Type | Min Radius | Jobsite Recommendation |
|---|---|---|
| 1/4" Flexible Drywall (Dry) | 24 inches (610 mm) | Double-layer required for fire and structural rigidity. |
| 1/4" Flexible Drywall (Wet/Steamed) | 12 inches (305 mm) | Moistened on back paper; fastened with fine drywall screws. |
| 3/8" Standard Drywall (Dry) | 72 inches (1828 mm) | Requires gradual bend along lengthwise grain. |
| 1/2" Standard Drywall (Dry) | 120 inches (3048 mm) | Only suitable for broad segmental room transitions. |
| 1/4" Kerfed MDF / Bendable Plywood | 8 inches (203 mm) | Ideal for tight radius soffits and arch window jamb returns. |
Use 3/4" CDX or birch plywood for arch ribs spaced max 16" on center. Skin the soffit with two layers of 1/4" bending ply. Support the arch formwork on double wooden folding wedges so the center can be struck smoothly after mortar cures.
Masonry arches must always have an odd number of voussoirs so the apex features a solid keystone rather than a vertical joint. Keep mortar joints between 1/8" (at intrados) and 3/8" (at extrados) or cut tapered voussoirs.
Structural Engineering Notice
IBC & IRC Code Compliance
This tool provides pure geometric, layout, and cut dimensions for architectural planning. It does not perform structural engineering load calculations.
- Never remove existing structural headers without an engineered beam.
- Frame the arch as an infill (cripple) structure below the engineered header.
- Consult a licensed structural engineer (PE) for masonry openings spanning over 6 feet.
Field Layout Methods & Interactive Drawing Simulators
Experience the 3 battle-tested techniques carpenters and masons use to transfer digital calculations onto physical plywood templates, drywall, and formwork.
The Gardener's Two-Pin String Method
Based on the geometric definition of an ellipse: the sum of distances from any point on the curve to the two focal points ($F_1$ and $F_2$) is always constant and equals the total opening span ($W$).
- Mark center line and draw springline (W = 48") and rise (H = 12").
- Focal pin distance: c = √(a² - b²) = √(24² - 12²) = 20.78".
- Drive two pins at distance c to left and right of center.
- Loop non-stretching string around pins, keeping pencil taut to sweep the fair curve.
Trade Applications & Cross-Section Blueprints
From rough framing to finished brickwork, see how our geometry engine prevents costly jobsite errors.
Doorway Stud Framing Cross-Section
Doorway & Cased Openings
Framing elliptical or segmental pass-throughs requires double 2x4 jack studs, plywood backing gussets, and accurate springline elevation alignment.
Brick & Masonry Openings
Always calculate an odd count of voussoir bricks to guarantee an apex keystone. For segmental brick arches, calculate skewback angle to ensure stability.
Arch Windows & Flexible Trim
Calculate exact arc lengths to order flexible polyurethane casing, pre-bent drywall corner bead, or laminated jamb extensions with zero waste.
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)- 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
Trade & Jobsite Tools
Tier 2 Trades (10)- 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
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 (4)- 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
- Segmental vs. Elliptical vs. Gothic →
Decision framework for ceiling height, framing & aesthetics
Frequently Asked Questions
Common carpentry, masonry, and geometric questions regarding arch calculations and construction layout.
Q1 How do you calculate the radius of an arch from width and height?
Using the geometric chord theorem: Radius R = (W² + 4H²) / (8H), where W is the total horizontal span opening and H is the vertical rise from the springline to the apex. For instance, an arch with a 48" span and a 12" rise yields: R = (48² + 4×12²) / (8×12) = (2304 + 576) / 96 = 30.00 inches.
Q2 What is the difference between a segmental arch and a semicircular arch?
A semicircular (Roman) arch is a full 180° half-circle where the rise is always exactly half the span (H = W/2) and the radius equals W/2. A segmental arch is only a partial segment of a circle (less than 180°), creating a lower, flatter profile where the center of curvature lies below the springline.
Q3 How do you lay out an arch when the radius is too large for a compass?
Use our offset-coordinate 'Mark-Out' table. Draw a horizontal baseline across your plywood or wall, mark 2" or 3" horizontal step intervals from the center point, and measure straight up to the calculated Y height at each step. Tap finish brads at each coordinate and bend a flexible batten strip across the pins to draw your cut line.
Q4 Why must masonry arches have an odd number of voussoirs?
Traditional structural brick and stone arches require an odd total number of voussoir wedges so that a single, symmetrical keystone sits precisely at the apex crown. An even number would create a vertical mortar joint directly down the center line, creating a structural fracture vulnerability under load.
Q5 How much drywall or flexible moulding do I need for an arched opening?
You need to measure the Arc Length (L), which is the curved perimeter along the intrados soffit. For circular and segmental arches, Arc Length L = Radius × Central Angle (in radians). For an opening with 48" span and 12" rise, the arc length is approximately 55.85 inches (4' 8"). Always add 10% for trim miter cuts.
Q6 Can standard drywall bend to any arch curve without cracking?
No. Standard 1/2" drywall has a dry bending radius limit of approximately 120 inches (10 feet). For tighter arches, use 1/4" flexible drywall (such as High Flex gypsum), which bends dry down to 24" radius, or wet down to a 12" radius. Alternatively, use kerfed MDF or double-layered bendable plywood for tight jambs.
Q7 How is an elliptical arch framed compared to a true circle?
True ellipses have continuously varying curvature (sharp radius at shoulders, flat radius at the crown). In framing carpentry, they are laid out using the Trammel of Archimedes method, the string-and-two-focus-pins method, or stepped coordinates using our calculated offset table. Multi-radius pseudo-ellipses use 3 or 5 circular tangent arcs to approximate the curve.