Moorish & Islamic Architecture (θ > 180°)

Horseshoe (Moorish) Arch Calculator

Calculate circular sweep angles (θ > 180°), maximum belly waist width (W_max), jamb neck constriction, and elevated center pivot height (h_c) for Moorish keyhole architecture.

Constricted Keyhole Profile Moorish
Constricted width
in
Must be > W/2
in
Historic Moorish Proportions: Keyhole
Horseshoe Keyhole Profile (θ > 180°)
MAX BELLY WAIST (W_max) 2R Diameter
56.00 in
+8.0" wider than opening
SWEEP ANGLE (θ) > 180° Sweep
218.0 °
3.805 radians
TOTAL APEX HEIGHT (H) R + h_c
42.42 in
3' 6 7/16" Total Rise
ARC LENGTH (L) Perimeter
106.5 in
8' 10 1/2" Curved Length
CENTER ELEVATION (h_c) 14.42 in above springline
NECK CONSTRICTION RATIO 85.7% of Max Width
INWARD INCLINE ANGLE 19.0° Inward Slope
Moorish Keyhole Simulator

Interactive Center Elevation & Neck Constriction Simulator

Adjust the radius slider to observe how expanding the upper circle beyond the springline increases the keyhole waist diameter and deepens the neck constriction.

28.0 Inches (θ = 218.0°)
Maximum Belly Diameter 56.00 inches Widest waist point
Subtended Sweep Angle 218.0° Sweep > 180° Moorish sweep
Center Height Above Springline 14.42 inches Pivot lift elevation
Field Layout Generator

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.

Coordinate Stepping Points (Centerline Outward)
Based on current Span: 48" • Rise: 12" • Radius: 30.00"
Step Interval:
Pt # Dist from Center (X) Height (Y) Fraction Drop
💡 Tip: Symmetrical — use identical Y heights for both left and right sides.

Live Plywood Scribing Simulator

Point 1 of 13
Plotted Offset (X): 0.00" (Center Apex)
Height Y: 12.000" (12") Drop: 0.000"
3/4" Plywood Blank Baseline X-Axis (Springline) Y: 12"

🔨 Carpentry Technique: Tap finish brads at coordinate marks. Bend a flexible batten against the pins to scribe a smooth curve.

Keyhole Trigonometry

Horseshoe Arch Mathematical Equations

The geometry of a circle whose center is raised above the horizontal impost line.

1. Center Lift & Sweep Angle
Center Elevation: h_c = √(R² − (W_spring / 2)²)
Sweep Angle: θ = 180° + 2 × arcsin(h_c / R)
Max Belly Width: W_max = 2 × R
Total Rise Height: H = R + h_c

The curve extends below the center line by an angle equal to arcsin(h_c / R) on both sides.

2. Perimeter & Proportions
Arc Length: L = R × θ (in radians)
Neck Constriction: Ratio = W_spring / W_max
Inward Incline: α = arcsin(h_c / R)
Jamb Width Offset: Overhang = (W_max − W_spring) / 2

Moorish master builders set $h_c$ between $R/3$ and $R/2$ for ideal classical aesthetic balance.

Step-by-Step Worked Numeric Example

Suppose you are framing an authentic Moorish horseshoe arch with a constricted jamb span of W = 36.0 inches, a total rise of H = 24.0 inches, and a swing radius of R = 20.0 inches.

Step 1: Center Elevation Rise − Radius = 24 − 20 Pivot is 4.0" above springline
Step 2: Shoulder Inset (2R − W) / 2 = (40 − 36)/2 2.0" constriction per side
Step 3: Total Sweep Angle 180° + 2×arcsin(4/20) θ = 203.07° (3.544 rad)
Step 4: Soffit Perimeter (L) 20.0 × 3.544 L = 70.88" (5' 10 7/8")
Carpentry & Framing Field Guide

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.

🛠 Recommended Jobsite Tools
Trammel Bar / Beam CompassFor swinging long-radius arcs onto plywood sheets with pinpoint accuracy
Flexible Batten / Fairing StripThin strip of PVC, vinyl, or straight-grain pine to bend across offset coordinates
Framing Square & LevelTo ensure a true horizontal springline baseline and plumb king studs
Plumb Laser or Laser LevelFor projecting centerlines and verifying vertical alignment across wide spans
Jigsaw / Flush-Trim RouterTo cut the master curved plywood rib template and duplicate identical gussets
Bar Clamps & Finish BradsTo secure flexible drywall or trim casing tightly along the curved soffit rib
01

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.

Set the elevation baseline and check opening squareness.
02

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.

Transfer calculated geometry to physical material.
03

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.

Create rigid curved framing ribs for the soffit.
04

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.

Build the structural framing sandwich.
05

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.

Install the curved soffit liner and finish casing.
Structural Mechanics & Physics

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.

Thrust Equation

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):

H_thrust = (Total Load × Span) / (8 × Rise)

Key Rule: Halving the rise height doubles the outward lateral force pushing against your king studs or masonry abutment piers!

Framing Reinforcement Rules
✓Span ≤ 48": Double 2x4 king studs each side
✓Span 48"–72": Triple king studs + 2x8 structural header
✓Span > 72": Engineered LVL header with structural tie straps
✓Masonry Arches: Skewback width must be ≥ 1.5× wall thickness

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).

Low Lateral Thrust

Semicircular (Roman) Arch

Lateral Thrust Ratio20% – 30%

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.

Vector: Primarily Vertical (Downwards)
Moderate Lateral Thrust

Gothic (Pointed) Arch

Lateral Thrust Ratio35% – 50%

The pointed apex divides the curve into steeper arcs, reducing horizontal outward vector forces compared to flatter round arches.

Vector: Steep Diagonal Vector
High Lateral Thrust

Segmental (Shallow) Arch

Lateral Thrust Ratio75% – 90%

As the rise decreases relative to the span, the thrust vector angles sharply outward, exerting immense lateral force on the supporting side walls.

Vector: Strong Outward Diagonal Kick
Severe Lateral Thrust

Jack (Flat) Lintel Arch

Lateral Thrust Ratio150% – 200%

Flat arches rely solely on wedge friction and severe horizontal thrust to hold the voussoirs aloft. Requires massive abutments or structural tie rods.

Vector: Extreme Horizontal Force
Trade Gotchas & Quality Control

Common Arch Doorway Framing & Casing Mistakes

Prevent unlevel springlines, irregular cripple stud lengths, and misaligned trim casing.

⚠️ DRYWALL & FRAMING

Exceeding Drywall Minimum Bending Radius

Common Mistake:

Attempting to force standard 1/2" drywall around tight curves (under 10 ft radius) dry, causing catastrophic gypsum core fracture and unsightly faceted creases.

Pro Trade Solution:

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).

⚠️ STRUCTURAL LOAD

Underestimating Lateral Horizontal Thrust

Common Mistake:

Treating shallow segmental or flat jack arches like standard post-and-beam lintels without reinforcing the side supporting piers or king studs.

Pro Trade Solution:

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.

⚠️ TRIM & FINISHING

Ordering Casing by Span Width Instead of Arc Length

Common Mistake:

Ordering flexible polyurethane or wood moulding based on the horizontal rough opening width ($W$), leaving the installer short by 15% to 57%.

Pro Trade Solution:

Always order moulding based on the true calculated Intrados Arc Length (L_intrados) plus a minimum 10% allowance for trimming springline plumb cuts.

⚠️ LEVELING & DATUM

Uneven Springline Heights Across Jambs

Common Mistake:

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.

Pro Trade Solution:

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

★ Pre-curve flexible drywall overnight around a cylindrical form before hanging to relax internal surface tension.
★ When making plywood rib templates, clamp both ribs together and sand simultaneously for identical profiles.
★ Always leave centering formwork and shoring wedges in place until masonry mortar achieves full 7-day compressive cure strength.
★ For flexible casing moulding, acclimatize the material to room temperature (70°F+) for 24 hours prior to bending.
Related Tools

Related Arch Shape Calculators

Explore complementary specialized geometry tools and layout solvers.

Expert Knowledge Base

Frequently Asked Questions

Common carpentry, masonry, and geometric questions regarding arch calculations and construction layout.

Q1 What is a horseshoe (Moorish / Keyhole) arch?

A horseshoe arch (also known as a Moorish or keyhole arch) is a circular arch whose curve extends beyond a 180° semicircle before curving inward toward the springline. The maximum width (waist diameter) is wider than the opening width between the jambs at the springing line.

Q2 How do you calculate the sweep angle and center elevation of a horseshoe arch?

The center of curvature (C) is elevated at a height (h_c) above the springline. The subtended sweep angle is θ = 180° + 2 × arcsin(h_c / R). For example, if Radius R = 30" and the center is raised h_c = 6" above the springline: θ = 180° + 2 × arcsin(6 / 30) = 180° + 2 × 11.54° = 203.07°.

Q3 What is the relationship between the springline width (W_spring) and maximum waist width (W_max)?

The maximum width is simply the circle diameter W_max = 2R. The constricted opening at the springline is W_spring = 2 × √(R² − h_c²).

Q4 Where is the horseshoe arch found in architectural history?

The horseshoe arch originated in pre-Islamic Visigothic Spain and Roman Syria, and became the quintessential hallmark of Islamic and Moorish architecture, celebrated in the Great Mosque of Córdoba and the Alhambra Palace in Granada.