How to Calculate Arch Radius from Width and Height
Master the geometric intersecting chords theorem to solve exact radii, sagitta heights, and pivot drop distances for woodworking, framing, and masonry templates.
The Master Chord Formula
Where W is the clear opening span and H is the vertical rise (sagitta) from the springline to the crown apex.
Full Pythagorean Proof
In any circular segment, draw a right triangle from the circle center to one end of the chord:
- Hypotenuse is the Radius
R - Horizontal leg is half the span
W / 2 - Vertical leg is
R − H - Apply Pythagorean theorem:
R² = (W/2)² + (R − H)² - Expand:
R² = W²/4 + R² − 2RH + H² - Cancel
R²:2RH = W²/4 + H² - Divide by
2H: R = W² / (8H) + H / 2
Worked Carpenter Example (36" × 12-inch)
For a doorway opening with span W = 36 inches and rise H = 12 inches:
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 (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
Trade & Jobsite Tools
Tier 2 Trades (12)- 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
- Pizza Oven Arch Calculator→
Firebrick counts & 63% door-to-dome thermodynamic ratio
- Barrel Vault Ceiling Calculator→
Curved ceiling framing ribs at 16" O.C. & drywall takeoffs
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 (6)- 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
- Types of Arches & Selection Guide→
Decision framework for ceiling height, framing & aesthetics
- How to Build a Brick Arch→
5-stage masonry guide on voussoir layout, centering & wedges
- How to Bend Drywall Around an Arch→
Dry flexing, wet sponging & back-kerfing for tight curves
- Calculate Arch Radius from Width & Height→
Pythagorean chord theorem proof & carpenter fraction math
Frequently Asked Questions
Common carpentry, masonry, and geometric questions regarding arch calculations and construction layout.
Q1 What is the formula to calculate arch radius from span and rise?
The formula is Radius R = (W² + 4H²) / (8H), where W is the horizontal span opening and H is the vertical crown rise.
Q2 Why does this formula work?
It derives directly from the Pythagorean theorem on the right triangle formed by half the chord (W/2), the vertical offset (R - H), and the hypotenuse radius (R): R² = (W/2)² + (R - H)².