Arch Calculator

Comprehensive Guide to Arch Geometry, Carpentry Framing, and Masonry Engineering

The architectural arch is one of civilization's foundational structural innovations, enabling builders to span extensive horizontal openings for doors, windows, entry gates, vaulted tunnels, and bridges without relying on high-tensile steel beams. Whether constructing a timber rough opening for a curved drywall passageway or laying a segmental brick arch over an outdoor fireplace, understanding circular arc geometry is crucial to achieving structural balance and aesthetic perfection.

When an arch is constructed improperly, the horizontal outward thrust exerted at the springline can rupture masonry piers, deform wood framing studs, or produce uneven mortar gaps along the outer circumference. To ensure seamless structural integrity, this free calculation engine applies rigorous chord-radius geometry, enabling craftsmen and design engineers to determine critical values such as true radius of curvature, intrados arc length, sagitta crown elevation, and voussoir brick quantities.

Mathematical Principles of Circular and Segmental Arches

A segmental circular arch represents a portion of a complete circular circle. Unlike an elliptical, parabolic, or catenary arch, every point along a true circular arch is equidistant from a single center point. The relationship between the clear horizontal width (the span or chord length, denoted as W or S) and the vertical height at the exact center (the rise or sagitta, denoted as H or R_rise) is derived directly from the intersecting chord theorem (Euclid's Elements, Book III, Proposition 35).

These formulas establish that the radius is governed entirely by the ratio of span to rise. When an arch is low and flat (a small rise over a large span), the radius expands exponentially. Conversely, when the rise equals half of the clear span (H = S / 2), the subtended central angle becomes exactly 180 degrees (π radians), producing a classic Roman semicircular arch where the radius is simply half of the span.

Essential Arch Terminology for Framers and Masons

Mastering structural arches requires a precise understanding of the traditional architectural lexicon used across job sites:

• Span (Chord): The clear horizontal distance between the inner faces of the vertical supporting jambs or abutments.
• Rise (Sagitta): The maximum vertical distance measured from the horizontal springline up to the highest point on the underside of the arch.
• Springline (Impost Level): The horizontal plane where the vertical jambs end and the curved trajectory of the arch begins.
• Intrados (Soffit): The inner curved concave surface of the arch opening. Arc length measurements taken along the intrados dictate rough framing cutouts and temporary centering forms.
• Extrados: The outer convex curved boundary of the arch ring. Because the extrados radius is larger than the intrados radius by the arch depth, its arc length is consistently longer.
• Crown: The highest central point along the arch curve.
• Haunches: The intermediate curved sections located between the crown and the skewbacks, which absorb substantial bending shear.
• Voussoirs: Wedge-shaped stones or tapered bricks that form the body of a masonry arch ring.
• Keystone: The central, wedge-shaped voussoir set into the crown that locks all adjacent units in place under mutual compression.
• Skewback: The angled or stepped seat formed on the supporting abutment from which the arch curve springs.

Step-by-Step Instructions: How to Use This Arch Calculator

This application features three specialized computation modes tailored to distinct job-site and drafting scenarios:

Mode 1: Segmental Arch from Known Span and Rise

Select this mode when framing arched doorways, windows, or pass-through openings where the rough width and desired drop or rise are predetermined:

1. Enter the clear horizontal span (e.g., 72 inches for a wide entryway).
2. Enter the vertical rise measured from the springline to the crown (e.g., 18 inches).
3. Enter the arch depth (framing thickness or wall width) if you need the volume and extrados boundary values.
4. Click Calculate Arch to generate the exact curvature radius, arc length along the soffit, chord angles, and an interactive, proportional vector SVG layout.

Mode 2: Arch Geometry from Known Radius and Span

Use this mode when matching existing historical architecture, restoration work, or prefabricated circular timber forms:

1. Input the clear opening span (chord length).
2. Enter the known radial dimension. Note that the radius must always be greater than or equal to half the span; otherwise, the geometry is physically impossible for a single circular arc.
3. Click Calculate Geometry to instantly solve for the resultant crown rise, central angle, and segmental area.

Mode 3: Masonry Brick and Voussoir Arch Estimator

This mode determines the exact quantity of masonry units and mortar joint taper required to construct a self-supporting brick arch:

1. Enter the clear span opening and rise.
2. Specify the standard brick thickness along its face (typically 2.25 inches for modular US brick).
3. Input the minimum mortar joint thickness at the intrados (commonly 3/8 inch or 10 mm).
4. Specify the radial depth of the arch ring (standard 8 inches for header bonds or 4 inches for rowlock soldier arches).
5. Click Calculate Brick Count to review required voussoirs, keystone width, and the calculated wedge expansion at the outer extrados.

Practical Guide: Framing a Timber and Drywall Arch

When framing a curved wood opening in modern residential construction, carpentry crews build curved plywood or Oriented Strand Board (OSB) headers called arch ribs:

• Laying Out the Radius with a Trammel Arm: Cut a 1x2 pine strip slightly longer than the calculated radius. Drill a small pilot hole at one end for a pencil point, and measure back along the strip to drill a pivot hole corresponding to the calculated radius. Secure the pivot point to your subfloor or layout bench, insert the pencil, and scribe a continuous, fluid arc across two layered sheets of 3/4-inch plywood.
• Cutting and Doubling Ribs: Cut along the pencil line using an electric jigsaw or bandsaw. Sand the cut smooth. Use this piece as a master template to route identical matching ribs. Two parallel plywood ribs separated by short 2x4 blocking (cripple blocks) form the load-bearing frame.
• Installing Flexible Backing: Fasten 1/4-inch bendable drywall or two layers of kerfed 3/8-inch drywall along the inner soffit (intrados). For outdoor or heavy applications, install 1/2-inch bendable plywood before applying drywall, stucco netting, or beadboard trim.

Engineering Thrust, Abutments, and Masonry Mechanics

Unlike a horizontal steel or timber header that exerts purely vertical downward forces onto jack studs, an arch converts gravity loads into diagonal lateral vectors. This phenomenon is known as horizontal thrust. As vertical weight presses down on the crown and haunches, the voussoirs compress tightly against one another, pushing outward against the skewbacks at the springline.

For an arch to remain stable throughout centuries of seasonal temperature shifts and structural settling, the line of thrust (the compression resultant vector) must remain entirely within the middle third of the arch ring depth. If the line of thrust exits the intrados, the bottom joints open and spall; if it breaches the extrados, the arch buckles upward. Adequate abutment mass, reinforced tie rods, or masonry buttresses are essential whenever flat segmental arches are integrated into multi-story brick facades.

For related structural estimations across your job site, consult our comprehensive framing tools including the Stair Calculator, check horizontal deadweight on foundations using our Concrete Slab Calculator, evaluate building envelope slope with the Roof Pitch Calculator, and estimate full-wall materials using the Brick Calculator.

Frequently Asked Questions (FAQ)