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Arch Calculator — Masonry Arch Dimensions & Materials

Calculate the arc length, rise, and material needed for brick or stone arches. Get precise measurements for semicircular and segmental arches.

About This Calculator

An arch is one of the oldest and most structurally efficient ways to span an opening in masonry construction. This arch calculator determines the arc length, radius, material volume, and estimated brick count for segmental and semicircular arches. Whether you are building a brick doorway, a stone garden arch, or a decorative window header, accurate calculations prevent material waste and ensure structural integrity.

The Formula Behind This Calculator

The calculator uses the geometric relationship between the span (chord length) and rise (sagitta) of a circular segment. First, the radius is derived from R = (span/2)² + rise² / (2 × rise). The half-angle subtended at the center is θ = arcsin(span / 2R). The arc length equals 2Rθ, and the segment area is 0.5 × R² × (2θ − sin(2θ)). Multiplying the segment area by the wall depth gives the arch volume. The brick estimate divides the arc length and wall depth by individual brick-plus-mortar unit dimensions.

Understanding the math helps you verify results and make better decisions for your project.

How to Use

  1. 1Enter the opening span — the total width the arch must bridge, measured in inches.
  2. 2Enter the arch rise — the vertical distance from the base of the opening to the highest point of the arch underside (the intrados).
  3. 3Enter the wall thickness (depth) that the arch will span through — typically 8 inches for a standard brick wall.
  4. 4Enter the brick or stone unit height and the mortar joint thickness for an accurate material estimate.

When to Use

  • Planning a brick arch over a doorway, window, or passageway in new construction.
  • Estimating materials for a garden or landscape stone arch feature.
  • Restoring or replicating a historical masonry arch and needing arc dimensions.
  • Designing a segmental arch for a retaining wall opening or bridge span.

Tips

  • A semicircular arch has a rise equal to half the span. A segmental arch has a rise less than half the span.
  • Always order 5–10% more bricks than calculated to account for cuts, breakage, and wastage at the arch edges.
  • The keystone (center brick) should be the last piece installed and must fit snugly to lock the arch in place.
  • Temporary formwork (a timber or plywood centering) must support the arch until mortar has fully cured — typically 7 to 28 days.

Understanding Arch Geometry and Structural Principles

Masonry arches work by transferring vertical loads into compressive forces that travel along the curve of the arch and down into the abutments on either side. The shape of the arch directly affects how those forces are distributed. A semicircular arch directs most of the force downward along the curve, while a flatter segmental arch exerts more horizontal thrust that must be resisted by the surrounding wall or by tie rods.

The three critical geometric dimensions — span, rise, and radius — are mathematically linked. Given any span and rise, the radius of the circle that the arch belongs to can be calculated precisely. This radius determines the curvature of every brick or stone voussoir (wedge-shaped arch stone) in the arch. Builders who understand this geometry can set out the arch on the ground or on formwork with a string line pinned at the center of the circle.

The intrados (inner curve) and extrados (outer curve) of the arch are concentric arcs. The thickness of the arch ring between them — sometimes called the arch barrel — must be proportioned to handle the load. For brick arches spanning up to 6 feet, a ring depth of one brick (4 inches nominal) is typical. Wider spans or heavier loads require a deeper ring, often calculated alongside the concrete slab thickness that may serve as the supporting lintel above.

Types of Masonry Arches in Construction

The semicircular or Roman arch is the most recognizable form, rising exactly half the span. Roman builders used this shape extensively in aqueducts, bridges, and doorways because it is simple to lay out and distributes weight efficiently. When the rise is significantly less than half the span, the result is a segmental arch — the most common type in modern brick veneer construction for window and door headers.

Other traditional forms include the jack arch (almost flat, with a minimal rise for drainage), the horseshoe arch (rise exceeds half the span, curving inward below the spring line), and the Gothic or pointed arch formed from two circular arcs. Each has different thrust characteristics. The jack arch in particular pushes outward significantly and almost always needs a steel angle or Rebar calculator lintel behind it for support.

Segmental arches dominate residential brickwork because they combine a graceful curve with manageable horizontal thrust. Most building codes allow segmental arches with a rise-to-span ratio of 0.15 or greater without engineered lintels, provided the masonry on either side has sufficient mass to resist the thrust. When in doubt, a structural engineer should verify the design — especially for double-wythe or cavity wall arches.

Materials and Brick Selection for Arches

Not every brick is suitable for arch construction. Voussoirs — the wedge-shaped bricks or stones that form the arch — must be able to withstand high compressive forces along the curved axis. Modular clay bricks with a compressive strength of at least 3,000 psi are standard. Engineering bricks rated at 8,000+ psi are preferred for heavy-load or deep arches. The total number of units needed can be estimated using the brick calculator alongside this arch calculator.

For stone arches, the material must be durable and free of laminations that could split under compression. Limestone, sandstone, and granite are traditional choices. Rubble stone arches use irregular stones with thick mortar joints, while ashlar arches use precisely cut voussoirs with thin joints. The stone choice affects the visual character and the structural capacity of the arch.

Mortar for arch construction should be Type N (medium strength) or Type S (high strength) for most above-grade applications. The mortar must be workable enough to fill tapered joints but stiff enough not to slump. A common mix is 1 part Portland cement, 1 part lime, and 6 parts sand by volume. Proper mortar coverage is essential for load transfer between voussoirs.

Building and Centering (Formwork) for Arches

Every masonry arch requires temporary support called centering or formwork during construction. The centering holds the arch in its correct shape until the mortar has cured enough to support its own weight and any superimposed loads. For small arches, a simple plywood template cut to the intrados curve works well. Larger arches may need a framed timber center with adjustable props.

The centering must be built to the exact curvature of the intrados and must be rigid enough not to deflect under the weight of wet masonry. Even a small settlement during construction can crack the arch. Builders typically leave the centering in place for at least 7 days, and up to 28 days for large or heavily loaded arches. This is similar in principle to the formwork timeline used when pouring a excavation depth.

Striking (removing) the centering is a critical moment. It should be done gradually — lowering the props evenly — so the arch begins to carry its own weight symmetrically. A sudden release can cause uneven loading and cracking. After the centering is removed, the arch should be inspected for any movement or joint separation before loads are applied.

Estimating Material Quantities and Costs

The arch calculator provides an estimated brick count based on the arc length, wall depth, and unit dimensions. This count represents the voussoirs in the arch ring itself. It does not include the abutment bricks below the spring line, the spandrel masonry above the arch, or any backup wall. For a complete wall estimate, combine this tool with a general lumber framing estimate to capture the full picture.

Material costs for an arch include the bricks or stone, mortar, any steel lintel reinforcement, and the timber or plywood for centering. Specialty tapered bricks for tight-radius arches can cost 2–3 times more than standard modular bricks because they are made to order. For budget arches, standard bricks can be laid with wedge-shaped mortar joints — the perpends are wider at the extrados and narrower at the intrados.

Waste factors for arch construction run higher than straight wall work. Budget 10% extra for breakage and cutting on semicircular arches, and up to 15% for tight-radius or segmental arches where more trimming is needed. The Concrete Block calculator needed per brick is also slightly higher in arches because of the tapered joints. A good rule of thumb is 0.4 cubic feet of mortar per 100 bricks in an arch versus 0.3 cubic feet in straight wall work.

Common Mistakes and How to Avoid Them

One of the most frequent errors in arch construction is getting the geometry wrong at the layout stage. If the centering does not match the true circle defined by the span and rise, the voussoirs will not fit properly and joints will vary in thickness. Always verify the radius by measuring from the center point to several positions along the intrados curve before laying any brick. This geometric check is as fundamental as verifying the roof pitch before cutting rafters.

Another common mistake is insufficient abutment mass. The horizontal thrust of the arch must be resisted by the masonry (or other structural elements) on either side. If the wall above the spring line is too short or too light, the arch will push the walls outward and crack. A rule of thumb is that the abutment height above the spring line should be at least 60% of the rise for segmental arches and 80% for semicircular arches.

Inadequate curing before loading is another pitfall. Applying roof loads or floor loads to a freshly built arch before the mortar has reached adequate strength can cause permanent deformation. Protect new arches from freezing temperatures and rapid drying for at least 72 hours. In hot weather, wet-curing with burlap or plastic sheeting prevents mortar from drying too fast.

Arch Design Standards and Code Considerations

In the United States, masonry arch design is governed by TMS 402/ACI 530/ASCE 5 (Building Code Requirements for Masonry Structures). This standard provides guidelines for arch geometry, minimum ring depth, allowable compressive stresses, and thrust resistance. For residential veneer arches, many jurisdictions accept prescriptive designs that meet minimum rise-to-span ratios without requiring a structural engineer's stamp.

The International Residential Code (IRC) addresses arches in Section R606, specifying minimum wall thickness, bond patterns, and support requirements. For arches spanning more than 6 feet or carrying concentrated loads, the IRC typically requires engineered design. These requirements align with the structural calculations used in wall framing and other load-bearing elements.

When arches are part of a seismic or high-wind design category, additional reinforcement may be required. Horizontal joint reinforcement in the arch ring and the abutment courses helps resist tensile forces from lateral loads. In some cases, stainless steel helical ties or wire reinforcement is specified to tie the arch to the backup wall. Always check local amendments to the model codes, as requirements can vary by jurisdiction.

Restoring and Repairing Existing Masonry Arches

Historic masonry arches can last centuries but may develop problems over time: mortar joint deterioration, brick spalling, cracking from foundation movement, or bowing from insufficient thrust resistance. Before repairing an existing arch, assess the underlying cause. Repointing cracked mortar joints without addressing the thrust problem will not prevent recurrence. Structural monitoring may be needed, similar to assessing whether an existing Roofing calculator can support added loads.

Repointing an arch requires removing deteriorated mortar to a depth of at least twice the joint width, then packing new mortar in layers. The mortar should match the original in strength, color, and joint profile. Using mortar that is too strong (too much Portland cement) can damage historic soft bricks by preventing natural movement. Lime-based mortars are often the correct choice for pre-1930 arches.

For severely damaged arches, a full rebuild may be necessary. The process involves documenting the existing geometry, carefully dismantling the arch brick by brick, rebuilding the centering, and relaying the arch with new or salvaged bricks. Dismantling should proceed symmetrically from both sides toward the keystone to prevent unbalanced collapse. Photography and dimensioned sketches of the original should be made before any demolition begins.

FAQ

What is the difference between a semicircular arch and a segmental arch?

A semicircular arch rises exactly half the span, forming a full half-circle with a radius equal to half the span. A segmental arch has a rise less than half the span, forming a flatter curve that is just a segment of a circle. Segmental arches are common in modern brickwork because they exert less horizontal thrust.

How do I know if my rise is too shallow for a stable arch?

As a general rule, the rise should be at least 15–20% of the span for structural stability. Very flat arches (rise less than 10% of span) rely heavily on the surrounding masonry and may require steel lintels or reinforced backup. Consult a structural engineer for spans over 6 feet or rises below 15% of span.

What brick size should I enter for standard modular bricks?

Standard modular bricks are 2.25 inches tall (nominal). With a 3/8-inch mortar joint, the unit height becomes 2.625 inches. Enter 2.25 for brick height and 0.375 for mortar joint. For king-size bricks, use 2.75 inches tall; for engineer bricks, use 2.75 inches.

Does this calculator work for Gothic (pointed) arches?

This calculator is designed for circular segment arches, which include semicircular, segmental, and horseshoe arches. Gothic or pointed arches use a different geometry (two circular arcs meeting at a point) and require separate calculations.

How much mortar do I need for the arch beyond what is in the joints?

The brick estimate includes mortar in the bed and perpend joints. For a typical arch with 3/8-inch joints, plan on roughly 0.3 to 0.5 cubic feet of mortar per 100 bricks. Mix slightly more to account for waste and filling the arch backing.

Can I use this for stone ( rubble) arches as well?

Yes. For stone arches, enter the average stone height and a typical mortar joint thickness (often 0.5 to 1 inch for rubble). The brick count becomes a rough stone estimate. Actual stone count varies with the irregularity of the material.

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