Arch Design — Radius, Arc Length & Material Takeoff
Calculate arch geometry and material quantities for semicircular, segmental, elliptical, Gothic, Tudor, and flat arches. Used in masonry, restoration, and structural design. Supports brick, stone, concrete, and wood with mortar estimates.
Why This Construction Metric Matters
Why: Correct arch geometry ensures structural stability and aesthetic proportion. Wrong radius causes thrust problems; wrong material quantities cause cost overruns and delays. Abutments must resist horizontal thrust.
How: Semicircular: R = Span/2, Arc = πR. Segmental: R = (Span²/8×Rise) + Rise/2. Gothic: R = √(Rise²+(Span/2)²). Area and volume drive material takeoff; voussoirs = arc length ÷ (thickness + joint).
- ●Support formwork until arch is complete and mortar cures.
- ●Keystone locks the arch; install last.
- ●Abutments must resist horizontal thrust.
- ●Order 10% extra for cuts and waste.
Compact Examples
Semicircular Arch
Perfect half-circle arch with rise equal to half the span
📏 Dimensions
🧱 Material & Construction
Brick
Standard clay brick, traditional arch material
Typical Uses: Fireplaces, Doorways, Windows, Decorative arches
Planning estimates only. Verify with a licensed engineer or contractor before construction.
📐 Construction Industry Facts
Semicircular arch: rise = span/2, most stable design.
— Masonry
Segmental arch: rise < span/2, common for doorways.
— Structural
Brick ~120 lb/ft³, stone ~165, concrete ~150.
— ACI
Voussoirs = wedge blocks; keystone at top.
— BIA
📋 Key Takeaways
- • Semicircular: R = Span/2, Arc = πR, Rise = Span/2
- • Segmental: R = (Span²/8×Rise) + (Rise/2) | Lower rise, common for doorways
- • Gothic: R = √(Rise²+(Span/2)²) | Pointed, high vertical load
- • Material = Area × Width × density | Voussoirs = segments
Did You Know?
Semicircular: rise = span/2. Most stable, classic Roman design.
Source: Masonry
Segmental: rise < span/2. Common in bridges, doorways.
Source: Structural
Elliptical: elegant curve. Ramanujan approx for arc length.
Source: Geometry
Gothic: two arcs meet at apex. Less horizontal thrust.
Source: Architecture
Brick ~120 lb/ft³, stone ~165, concrete ~150.
Source: Materials
Voussoirs = wedge blocks. Keystone at top.
Source: Construction
How Arch Dimensions Are Calculated
Radius and arc depend on arch type. Semicircular: R=Span/2. Segmental: R from rise and span. Area → volume → material.
Semicircular
R = Span/2, Arc = πR
Segmental
R = (Span²/8×Rise) + Rise/2
Expert Tips
Formwork
Support until arch is complete.
Keystone
Last block locks the arch.
Abutments
Must resist horizontal thrust.
Order Extra
10% waste for cuts.
Arch Type Comparison
| Type | Rise Ratio | Uses |
|---|---|---|
| ⭕ Semicircular Arch | 50% | Roman architecture |
| ◐ Segmental Arch | 15% | Bridges |
| ⬭ Elliptical Arch | 30% | Grand entrances |
| 🔺 Gothic (Pointed) Arch | 60% | Cathedrals |
| 🏛️ Tudor Arch | 40% | Tudor architecture |
| ▭ Flat Arch | 5% | Lintels |
Frequently Asked Questions
Semicircular arch formula?
R = Span/2. Arc = πR. Rise = Span/2. Most stable design.
Segmental vs semicircular?
Segmental has lower rise. R = (Span²/8×Rise) + Rise/2. Common for doorways.
What are voussoirs?
Wedge-shaped blocks forming the arch. Keystone is the center top block that locks the arch.
Brick density for arches?
~120 lb/ft³. Stone ~165. Concrete ~150. Used for material takeoff.
Formwork needed?
Yes, support until arch is complete and mortar cures. Remove carefully.
Horizontal thrust?
Arches push outward at the spring line. Abutments must resist this thrust.
Key Numbers
📚 Sources
⚠️ Disclaimer: Estimates for planning. Structural arches require engineering. Verify with professional.
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