Gothic (Pointed) Arch Calculator
Calculate crown apex height, equilateral vs. lancet arc radii (R), apex miter cut angles (α), and perimeter soffit lengths for cathedral doors, church windows, and revival architecture.
Interactive Gothic Steepness & Miter Simulator
Adjust the radius-to-span ratio ($R/W$) to dynamically see how the apex peak steepens from a flat Tudor-like Drop arch to an ultra-tall soaring Gothic Lancet arch, calculating the exact miter saw angle for wood casing.
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.
Gothic Arch Trigonometry & Apex Math
Constructed from two congruent circular arcs whose center points are separated by distance $(2R - W)$ along the baseline.
Set your compound miter saw to angle α to cut the peak joints of the curved trim casing.
In an equilateral arch, the center pivots sit exactly on the opposite springing points (Spread = $W$).
Step-by-Step Worked Numeric Example
Suppose you are building a custom Equilateral Gothic entryway door with a clear opening span of W = 48.0 inches ($R = W = 48.0"$).
Standard Gothic Pointed Arch Opening Sizes
Equilateral and lancet dimensions, apex crown heights, subtended angles, and leaf perimeter lengths for standard architectural openings.
| Opening Purpose | Span (W) | Apex Rise (H) | Radius (R) | Half Arc (L) | Total Perimeter | Apex Cut | Recommended Substrate |
|---|---|---|---|---|---|---|---|
| Small Church / Chapel Window | 24.0" | 20.78" | R = 24.0" | Leaf = 25.13" | 50.27" Total | 30.0° Miter | Equilateral Gothic Frame |
| Standard Entryway Doorway | 36.0" | 31.18" | R = 36.0" | Leaf = 37.70" | 75.40" Total | 30.0° Miter | Tudor / Gothic Timber Jamb |
| Double Portal Entrance | 48.0" | 41.57" | R = 48.0" | Leaf = 50.27" | 100.53" Total | 30.0° Miter | CDX Plywood Ribs & Flex Poly |
| Great Hall / Church Sanctuary | 72.0" | 62.35" | R = 72.0" | Leaf = 75.40" | 150.80" Total | 30.0° Miter | Structural Glulam / Masonry |
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 Gothic Pointed Arch Layout & Miter Mistakes
Prevent apex alignment skew, incorrect miter saw angles, and casing shortfall along pointed curves.
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 an Equilateral Gothic arch?
An Equilateral Gothic arch is a two-centered pointed arch where the radius of both intersecting arcs is exactly equal to the span (R = W). The center of each arc sits directly on the opposite springing point. For an opening of span W, the apex height is H = W × (√3 / 2) ≈ 0.866 × W, and the two arcs meet at a 60° angle at the peak (requiring a 30° miter cut).
Q2 What is the difference between an Equilateral, Lancet, and Drop Gothic arch?
The three Gothic arch types are defined by the ratio of radius R to span W: (1) Drop / Blunt Arch: R < W (flatter, broader peak); (2) Equilateral Arch: R = W (classic proportions); (3) Lancet Arch: R > W (tall, narrow, steeply pointed peak typical of High Gothic cathedral windows).
Q3 How do you calculate the apex miter cut angle for Gothic arch casing moulding?
The miter cut angle at the peak where the left and right trim mouldings meet is Miter Angle = arcsin((R - W/2) / R). For a standard equilateral arch (R = W), Miter Angle = arcsin((W - W/2)/W) = arcsin(0.5) = 30.0° from the vertical centerline.
Q4 Why were Gothic pointed arches structurally superior in medieval cathedrals?
By pointing the arch apex upward, the compressive load line is directed much more steeply downward toward the ground rather than pushing outward laterally. This reduced outward thrust allowed medieval masons to build soaring cathedral walls with massive stained glass windows and thinner stone buttresses.