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.
Jobsite Guide: How to Find the Focal Points of an Elliptical Arch
To locate the two focal points (pins) of an elliptical arch for the string layout method:
- Determine semi-major axis (a): Measure half of the total horizontal span (
Span / 2). - Determine semi-minor axis (b): Measure the vertical rise (
Rise Height). - Calculate focal distance (c): Apply the ellipse focal formula: c = √(a² − b²).
- Drive focal pins: Measure distance
cleft and right from the center point along the baseline, and drive a screw or nail into each mark. - Set string loop: Tie a non-stretch cord around both pins so that when pulled taut with a pencil to the crown apex, it forms a triangle of length
2a + 2c.
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.
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.
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 ($2a$).
Position two finish nails at distance $c$ on either side of the centerline along the baseline.
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").
Standard Elliptical Arch Opening Sizes & Foci Table
Pre-calculated focal pin distances (c), two-pin string loop perimeters, and crown rise dimensions for standard residential cased openings.
| Opening Purpose | Span (W) | Rise (b) | Semi-Axes (a, b) | Focal Dist (c) | String Loop | Soffit Arc | Recommended Substrate |
|---|---|---|---|---|---|---|---|
| Standard Passage Arch | 36.0" | 12.0" | a = 18.0", b = 12.0" | c = 13.42" | Loop = 62.8" | 47.88" | 1/4" High-Flex Drywall |
| Double Pocket Doorway | 48.0" | 15.0" | a = 24.0", b = 15.0" | c = 18.73" | Loop = 85.5" | 62.73" | Double 1/4" High-Flex |
| Dining Room Cased Opening | 60.0" | 18.0" | a = 30.0", b = 18.0" | c = 24.00" | Loop = 108.0" | 77.74" | 1/4" Bendable Plywood |
| Living Room Great Arch | 72.0" | 18.0" | a = 36.0", b = 18.0" | c = 31.18" | Loop = 134.4" | 89.65" | Kerfed 3/4" MDF Jamb |
| Grand Gallery Hallway | 96.0" | 24.0" | a = 48.0", b = 24.0" | c = 41.57" | Loop = 179.1" | 119.53" | Wiggle Board & Poly Trim |
How to Lay Out & Build an Elliptical Archway
Step-by-step master carpentry guide for setting focal pins, tying string loops, cutting plywood ribs, and installing flexible trim.
Establish Major and Minor Axes on Template Sheet
Lay out a full sheet of 3/4" plywood. Snap horizontal major axis ($2a = W$). At exact midpoint, draw perpendicular minor axis line rising to crown height ($b = H$).
Locate Focal Points & Anchor Guide Pins
Compute focal distance $c = sqrt{a^2 - b^2}$. Measure distance $c$ from center out toward both left and right springline ends. Drive two stout finish screws or pins exactly at these focal points.
Tie Precision String Loop & Scribe Semi-Ellipse
Tie a closed loop of non-stretch string with total perimeter length $Loop = 2a + 2c$. Loop string around both focal pins. Place pencil inside loop, pull taut, and glide smoothly from left springline through crown apex to right springline.
Cut Master Plywood Rib & Router Duplicates
Cut along the scribed line with a jigsaw. Sand smooth. Use a flush-trim router bit with a guide bearing to produce matching front and back wall ribs.
Mount Framing Sandwich & Fit Flexible Trim
Fasten 2x4 infill blocking between ribs at 6" on center. Secure arch unit into rough opening. Finish with two plies of 1/4" High-Flex drywall or flexible polyurethane trim ordered to the calculated Ramanujan 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 Elliptical Arch String & Layout Mistakes
Avoid string stretch distortion, incorrect focal pin placement, and drywall fracture at sharp shoulders.
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 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.