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Segmental Brick Arch Math

Calculate the striking radius, arc angle, and intrados arc length for structural and veneer masonry segmental arches. Covers formwork sizing, brick count estimation, voussoir joint taper, mortar selection, and minimum rise-to-span ratios for structural arches.

Opening Geometry

Clear Arch Span (Width)

inches

The physical inside width between the vertical springers

Arch Rise (Camber)

inches

The vertical distance from the spring line to the apex

Arch Centering & Math

Arch Radius (Pivot Point)

60.00inches

From apex down to striking center

Arc Angle

73.7°

Intrados Arc Length

77.22inches

Estimator Tool

Divide length by masonry unit width

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What is Masonry Geometry: The Segmental Arch — Compression Structures, Formwork Sizing & Voussoir Layout?

A segmental arch is a circular arc of less than 180° that bridges a masonry opening (window, door, fireplace) using small units (bricks, stones, or concrete blocks) laid in continuous compression. Unlike a lintel beam that resists loads through bending (tension on the bottom face), an arch converts vertical loads into diagonal compression along its curved path, directing forces outward and downward into the flanking wall mass (abutments). The arch operates entirely in compression — the weakest point of any masonry arch is NOT the crown, but the mortar joints between voussoirs. When properly designed with adequate rise-to-span ratio and sufficient abutment mass, a segmental arch can span openings for centuries with no steel reinforcement. The geometry calculator determines the three critical dimensions needed to build the wooden centering formwork: the striking radius (R), the total sweep angle, and the intrados arc length.

Mathematical Foundation

Arch Radius (Striking Center)

R = \frac{\text{Rise}^2 + (\frac{\text{Span}}{2})^2}{2 \times \text{Rise}}

R

= The radius from the structural striking center up to the apex of the arch (in). The striking center is the pivot point from which the mason swings a string line, trammel bar, or plywood template to scribe the exact curvature onto the wooden centering formwork. For a semicircular arch, R = Span/2 and the center falls exactly on the spring line. For a segmental arch (less than 180°), the center falls BELOW the spring line. The lower the rise relative to the span, the longer the radius — and the further below the spring line the center drops. A flat segmental arch (e.g., 4″ rise on a 72″ span) can have R > 100″, requiring a trammel bar or carefully marked plywood template because a simple string line is too long to hold accurately. For restoration projects, the original radius can be reverse-engineered by measuring the span and rise of the existing arch, even if partially collapsed.

Rise

= The vertical camber distance from the spring line (top of the skewback / springing block) to the crown (highest point of the intrados curve) in inches. Rise determines the arch’s visual profile and, critically, the magnitude of horizontal thrust at the abutments. A minimum rise/span ratio of 1″ per 12″ of span (roughly 1:12 or 8.3%) is the general masonry rule for structural segmental arches. At a 1:12 ratio (e.g., 6″ rise on 72″ span), thrust is manageable. Below 1:18 ratio, horizontal thrust becomes extreme and the arch approaches jack arch behavior — requiring steel reinforcement or a structural steel angle concealed behind the masonry veneer.

Span

= The horizontal clear opening between the inner faces of the two springer blocks (abutments) in inches. Span is measured from the inside face of the left skewback to the inside face of the right skewback — NOT from the outside of the supporting wall pier. The span directly determines the magnitude of the structural load the arch must support (self-weight plus any superimposed masonry above the arch). Building codes typically limit unreinforced segmental brick arches to spans of 6–8 feet for veneer and 4–6 feet for structural load-bearing applications. Exceeding these limits requires engineering analysis for thrust resolution and abutment capacity verification.

Intrados Arc Length

L = 2 \pi R \times \left(\frac{\text{Angle}}{360}\right)

L

= The total length of the inner curve of the arch in inches — the surface that the wooden centering formwork must exactly match. This is the dimension the carpenter needs to cut the plywood template and the mason needs to lay out brick positions. Divide L by (brick width + one mortar joint thickness) to estimate total brick count. Example: L = 74.3″ / 2.5″ (brick + joint) = 29.7 → 30 bricks required. Always round UP and add 2–3 extra bricks for cuts and breakage. The intrados arc length is always longer than the span (a common mistake is ordering bricks based on span alone, resulting in a shortage).

Angle

= The total central angle subtended by the arch arc in degrees. For a semicircular arch: Angle = 180°. For a segmental arch: Angle = 2 × arcsin(half-span / R). A lower rise produces a smaller angle and a flatter arch. Typical decorative segmental arches range from 30° to 90°. Below 30°, the arch is very flat and thrust loads become extreme. Above 120°, the arch approaches semicircular and can be treated with simpler semicircular formwork.

Laws & Principles

The Jack Arch vs Segmental Arch structural distinction: A flat jack arch has zero rise and relies entirely on friction between perfectly cut, tapered voussoirs and enormous horizontal thrust resistance from the abutments. A segmental arch introduces a structural camber (rise), which progressively shifts the load vector from purely horizontal thrust (dangerous) into a combination of vertical compression and manageable horizontal thrust. At a rise-to-span ratio of 1:4 (Span/4 rise), approximately 70% of the load is carried in vertical compression and only 30% as horizontal thrust. At 1:12, thrust dominates at roughly 60%. The minimum rise for a structurally reliable segmental arch: 1 inch per foot of span (1:12). Below this, the arch requires concealed structural steel support (typically a steel angle or flat bar behind the masonry) and functions as a decorative veneer rather than a true structural arch.
The Striking Center and formwork construction: The striking center is the radial pivot point from which the carpenter scribes the arch curvature. For a semicircular arch (180°), this point sits on the spring line (level with the bottom of the arch). For a segmental arch, this point drops BELOW the spring line by a distance of (R − Rise). The formwork construction procedure: (1) Establish the spring line on both abutments (level, plumb, at the specified height). (2) Find the striking center by measuring (R − Rise) down from the spring line at the midpoint of the span. (3) Drive a nail at the striking center and swing a trammel bar or string line of length R to scribe the intrados curve onto plywood. (4) Cut two plywood ribs to this curve. (5) Separate the ribs by the wall thickness using spacer blocks. (6) Skin the top surface with thin plywood (1/4″ luan) bent over the ribs to create a smooth, continuous formwork surface for laying bricks against. (7) Support the entire centering on adjustable posts (shores) so it can be eased (lowered) gradually after the mortar cures, allowing the arch to settle into compression under its own weight without sudden shock loading.
Thrust resolution and abutment design: Every segmental arch generates horizontal thrust at the springers (H_thrust ≈ w × Span² / (8 × Rise), where w is the total distributed load per unit length). This thrust must be resisted by the adjacent wall mass acting as abutments. If the abutments are too light or too narrow, the arch pushes them outward and collapses. Rules of thumb: (1) Minimum abutment width = 1× the arch span or 16″, whichever is greater, for single-wythe veneer arches up to 5′ span. (2) For wider spans or load-bearing arches: refer to engineering tables or TMS 402 (Building Code Requirements for Masonry Structures). (3) If the arch sits near a corner or wall termination, horizontal thrust has no continuous wall mass to resist it — a reinforced bond beam or steel tie rod is required at the spring line level. (4) In restoration work: evidence of abutment failure includes horizontal cracking at the spring line level a few courses below the arch, visible lean in the wall above the arch, and gap opening between arch voussoirs at the crown.
Mortar joint taper and voussoir geometry: In a segmental arch, every mortar joint must taper (wider at the extrados, narrower at the intrados) so the bricks align radially toward the striking center. The joint taper angle = (brick depth) / R radians. Standard brick (3-5/8″ deep): at R = 85″, joint taper angle = 3.625/85 = 0.0426 rad = 2.4°. Mortar joints should be 3/8″ minimum at the intrados (narrow face) and increase to a maximum of 3/4″ at the extrados (wide face). If the calculated extrados joint exceeds 3/4″, the bricks must be cut (tapered) on a wet saw to reduce the required mortar joint. Alternatively, use purpose-made arch bricks (voussoir bricks) that are manufactured with a built-in taper. Type S mortar (minimum 1,800 PSI compressive) is required for all structural arches; Type N (750 PSI) is acceptable only for non-structural decorative veneer arches with concealed steel support.

Step-by-Step Example Walkthrough

" A mason needs to build the wooden centering formwork to hold a decorative brick arch over a 6-foot wide multi-pane window. The architect has specified a subtle 8-inch rise at the center apex. "

1. Identify inputs: Span = 72 inches. Rise = 8 inches.
2. Find half-span: 72 ÷ 2 = 36 inches.
3. Apply Radius formula: (8² + 36²) ÷ (2 × 8) = (64 + 1296) ÷ 16 = 1360 ÷ 16 = 85 inches.
4. Calculate Arc Angle: 2 × arcsin(36 / 85) = 2 × 0.437 rads = 0.875 rads = 50.1°.
5. Find Arc Length: 2 × π × 85 × (50.1 / 360) = 534 × 0.139 = 74.3 inches.
6. Brick count: 74.3″ / 2.5″ (standard brick + 3/8″ joint) = 29.7 → 30 bricks. Order 33 (10% waste).
7. Joint taper: 3.625 / 85 = 0.043 rad = 2.4° per brick. Intrados joint: 3/8″. Extrados joint: ≈ 9/16″. Both within acceptable range — no brick cutting needed.

Final Result: Striking radius = 85″ (trammel bar length). Arc sweep = 50.1°. Formwork contact surface = 74.3″. 30 bricks required (order 33). Joint taper = 2.4° per brick (3/8″ intrados to 9/16″ extrados). Striking center drops below spring line by: 85 − 8 = 77″.

Quick Answer: How do you calculate a segmental brick arch radius and formwork dimensions?

R = (Rise² + (Span/2)²) / (2 × Rise). Then Arc Length = 2πR × (Angle/360). Example: 72″ span, 8″ rise: R = (64 + 1,296) / 16 = 85″ striking radius. Arc angle = 50.1°. Intrados arc = 74.3″ (this is the formwork contact length and the dimension for brick count estimation). The striking center drops 77″ below the spring line. Minimum rise: 1″ per foot of span (1:12 ratio) for structural segmental arches without concealed steel.

Masonry Arch Types & Rise-to-Span Guidelines

Rise-to-span ratio determines structural behavior, horizontal thrust, and whether the arch is self-supporting or requires concealed steel reinforcement.

Arch Type	Rise/Span Ratio	Arc Angle	Thrust Level	Steel Required?
Jack (flat)	0 (no rise)	0°	Extreme	Yes — always
Flat segmental	1:18 to 1:12	25–40°	Very high	Usually (veneer)
Standard segmental	1:12 to 1:6	40–80°	Moderate	No (if abutments adequate)
Deep segmental	1:6 to 1:3	80–130°	Low	No
Semicircular (Roman)	1:2 (Rise = Span/2)	180°	Minimal	No
Horizontal thrust ≈ w × Span² / (8 × Rise). Doubling the rise cuts thrust in half. A semicircular arch has the lowest thrust for a given span because it directs nearly all force vertically. Flat segmental arches are the most architecturally popular (subtle, elegant) but the most structurally demanding. Most residential “brick arches” over windows are flat segmental veneer with a concealed steel lintel angle doing the actual structural work.

Pro Tips & Common Brick Arch Mistakes

Do This

✓Dry-lay your bricks on the centering formwork before mortaring to verify even joint spacing across the entire arc — adjust brick positions to avoid a thin “sliver” joint at the crown (keystone). The most visible defect in a brick arch is an asymmetric or unusually thin joint at the crown caused by poor layout planning. Divide the intrados arc length by your (brick + joint) module to check if the total comes out to an even number. If it doesn’t: slightly open or close all joints uniformly across the arc (adding 1/32″ per joint across 30 joints = nearly 1″ of total adjustment). Always start laying from BOTH springers simultaneously toward the crown, and use the keystone (crown brick) as the final closing unit. The keystone joint should match the average joint width, not be narrower or wider. Mark brick positions on the formwork with pencil during dry-lay for accurate mortared placement.
✓Ease (lower) the wooden centering gradually over 48–72 hours after the mortar has cured — never strip formwork suddenly. When the centering is removed, the arch must transfer from being supported by the formwork to being self-supporting through compression. Sudden removal causes shock loading: the arch settles abruptly, and mortar joints that haven’t fully cured may crack or separate. The correct procedure: (1) Wait minimum 7 days for Type S mortar cure (72 hours for initial set, 7 days for 75% strength). (2) Ease the shores (adjustable posts supporting the centering) by 1/8″ per day over 3–4 days, allowing the arch to gradually pick up its own weight. (3) Monitor for any hairline cracking at the extrados during easing. Minor crown settling of 1/16″ to 1/8″ is normal and indicates healthy compression loading.

Avoid This

✗Don't estimate brick count from the span dimension alone — the intrados arc length is always longer than the span, and the difference grows with rise. For a 72″ span with 8″ rise: arc = 74.3″ (3.2% longer than span). At 12″ rise on 72″ span: arc = 77.8″ (8% longer). At a semicircular rise (36″ on 72″ span): arc = 113.1″ (57% longer!). Estimating brick count from span results in a shortage. Always use the calculator’s intrados arc length as the basis for brick ordering, then add 10% for cuts and breakage. For arches with tapered (cut) voussoirs, waste can reach 15–20% because each brick requires an angled wet-saw cut.
✗Don't build a segmental arch near a wall corner or termination without resolving the horizontal thrust — the arch will push the corner outward and fail. Segmental arches rely on the flanking wall mass acting as abutments to resist horizontal thrust. At a wall corner, there is no continuous wall on one side to resist the thrust. The arch pushes the corner out. Solutions: (1) Install a concealed steel tie rod (threaded rod with bearing plates) at the spring line level connecting the two springers through the wall. (2) Use a reinforced concrete or steel bond beam at the spring line. (3) If the arch is veneer only (decorative over a steel lintel), thrust is minimal and the concealed lintel absorbs any tendency to spread. Never build a structural segmental arch within 2× the span distance from a corner or free wall termination without engineering review.

Frequently Asked Questions

Why does the striking center fall below the spring line for a segmental arch?

Because a segmental arch is less than a semicircle. In a semicircular arch, the radius equals half the span and the center sits exactly on the spring line. In a segmental arch, the rise is less than half the span, so the arc is a “shallow slice” from a larger circle. The center of that larger circle must be further away (below) to produce a gentle curve. How far below? The distance is (R − Rise). For the example: R = 85″, Rise = 8″, so the center is 77″ below the spring line. This is important for the carpenter building the centering: the trammel pivot point must be physically accessible at this location (sometimes in the floor below the window, sometimes requiring a temporary post extension below the scaffold). If the center falls in an inaccessible location, the carpenter inscribes the arc using a plywood template calculated from the radius and arc angle instead of swinging a physical trammel.

Can I build a brick arch without wooden centering formwork?

Only in very specific circumstances — and generally not recommended. Traditional masons in some regions built small arches (< 24″ span) “freehand” by leaning bricks against each other with fast-setting mortar, but this requires extreme skill and limits the arch to shallow spans. For any arch over 24″ span, wooden centering is required because: (1) fresh mortar has virtually zero tensile strength — the unset arch is NOT a compression structure until the mortar cures and all voussoirs lock into compression against each other; (2) gravity will pull incompletely supported bricks out of alignment while the mortar is plastic; (3) the curvature must be geometrically precise for the compression line to follow the correct path through the voussoirs. Modern alternative: some masons use rigid foam (EPS or XPS) as centering formwork because it’s lightweight, easy to cut to curved profiles, and can be dissolved or collapsed after cure. However, foam lacks the stiffness to support heavy structural arches beyond a single wythe of veneer brick.

How do I calculate the mortar joint taper for each voussoir?

Each mortar joint in a segmental arch must taper (wider at the extrados, narrower at the intrados) so all joint lines radiate toward the striking center. Joint taper calculation: Taper per brick = brick depth × (intrados joint width / R). For a standard brick (3-5/8″ deep) with R = 85″ and 3/8″ intrados joint: extrados joint = intrados_joint + (brick_depth × intrados_joint / R) × 2 ≈ 3/8 + (3.625 × 0.375 / 85) × 2 ≈ 3/8″ + 1/32″ ≈ 13/32″. In practice: with a long radius (R > 60″), the taper is gentle and standard rectangular bricks produce acceptable joints (less than 3/4″ extrados). With a short radius (R < 30″; deep arches or small spans), the taper is dramatic and bricks must be tapered-cut on a wet saw or purpose-made tapered voussoir bricks used to keep the extrados joint within the 3/4″ maximum. Rule: if (brick_depth / R) > 0.08, the bricks likely need cutting.

What mortar type should I use for a structural brick arch?

Type S mortar (ASTM C270; minimum 1,800 PSI compressive strength) is required for all structural masonry arches. The arch operates in pure compression: every mortar joint must resist the compressive stress from the diagonal load path without crushing. Type S provides excellent compressive strength and good bond strength for the tapered wedge-shape joints in arch voussoirs. Type N (750 PSI) is acceptable ONLY for non-structural decorative veneer arches where a concealed steel lintel carries the actual structural load — the brick arch is purely cosmetic. Never use Type O (350 PSI) in any arch application; its low compressive strength cannot resist arch loads. For historical restoration: lime-based mortars (Type K or hydraulic lime) are used to match the original mortar profile, but these require significantly longer cure times (21–28 days before centering removal vs. 7 days for Portland-based Type S). Consult the project specifications and local building code for the required mortar type in your jurisdiction.