Masonry

Turning a Brick Arch Over an Opening

One number — the rise — settles what family the arch belongs to, how hard it pushes sideways, and the taper on every wedge you cut.
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Waiting on One Number at the Window Head

The reveals are up, the jambs are plumb, and the opening is closed off to its full width. On the boards there is a pack of soft facings, a sheet of ply nobody has shaped yet, and a gang with nothing to do until somebody says how high the arch is going to be. Span is not the argument — the brickwork already fixed it and a tape settles it in one measurement. Rise is the argument, and on a great many jobs it has never actually been decided: it exists as a curve drawn at 1:50, where the thickness of the pencil covers the difference between two entirely different pieces of work.

Nothing about the day can start until that number is real. Rise fixes the radius, and the radius fixes the angle each brick sweeps through, which fixes how many bricks are in the ring, which fixes whether the ring closes with a key at the crown or with a joint. Rise also fixes the taper — the difference between the width of a brick's face at the soffit and its width at the back — and the taper decides whether you are laying uncut bricks with fat joints, cutting wedges on a saw, or ordering purpose-made specials with a lead time measured in weeks. And rise fixes how hard the arch pushes sideways at the springing, which decides whether the jambs either side of the window are anything like adequate.

The classification is the shortest way to say all of that at once. Divide rise by span and the answer lands in one of three bands: about an eighth or less and the thing is a flat or jack arch, between roughly an eighth and a half it is segmental, and at exactly a half it is semicircular, because the springing points and the crown then sit on a circle whose centre is on the springing line. Those are not decorative labels. Each band is a different structural problem, a different setting-out method, and in practice a different quotation.

Put the measured span and the proposed rise in before anything else happens on the scaffold — the band the ratio lands in is what tells you which of the next six sections you are actually going to be doing.

The horizontal clear span of the arch opening.

The vertical height from the springline to the crown of the arch.

Rise-to-span ratio

0.125 (rise/span)

High confidence

Classification: Flat / jack arch.

Radius to strike the curve
4.25 ft
Centre point, below the springline
3.75 ft
Angle the arch subtends at that centre
56.14 °
Skewback, from horizontal
61.93 °
Length along the intrados
4.16 ft
4 ft6 in
Schematic, drawn to the proportions you entered — not to scale on screen.

What this calculation does not cover

  • This classifies a shape; it is not a structural check. It returns no thrust force, no abutment size and no load capacity, so it cannot tell you whether the arch, its supports or the wall above will stand. An arch over a structural opening needs a designer's check, not a ratio.
  • The bands describe circular-segment arches. Pointed, Gothic, Tudor, three-centred, elliptical and horseshoe shapes are not distinguished by rise over span, and a ratio much above 1/2 is not a semicircle at all — past a small margin over 1/2 no name is offered, so read the ratio itself against the geometry.
  • The set-out above is the CURVE, not the arch. Radius, centre, subtended angle, skewback and the length along the intrados are enough to strike the line and build the centring, and they are exact for a circular segment — but no voussoir is counted or sized here, because that needs the brick, the joint and the ring depth, none of which this page asks for. The brick arch wedge taper calculator takes those.
  • The centre point is given as a depth below the springline on the vertical centreline, which is where a trammel is pinned. On a flat or jack arch that depth is large — a 1.2 m (4 ft) span rising 150 mm (6 in) strikes from 1.125 m (3.7 ft) below the springline — so setting one out on site usually means a string and a pin in the floor rather than a beam compass, and the accuracy of the curve is the accuracy of that pin.
  • Ring depth, number of rings, brick and mortar type, and the load and backing carried above the arch are not inputs, and every one of them changes how a given shape behaves. Two arches with the same rise-to-span ratio can be sound and unsound.
  • The answer assumes rise is measured vertically from the springline to the underside of the crown, and span is the clear opening between abutments. Measuring rise from a sill or floor, or span to the outside of the jambs, shifts the ratio and can move it into a different band.

Thrust Goes Wherever the Rise Isn't

An arch works by turning vertical load into an inclined force running down through the ring to the springing. Only part of that force is vertical; the rest pushes outwards, and the flatter the arch the larger that outward share becomes. The relationship is easiest to see in the classical result for a uniformly loaded parabolic arch, where the horizontal thrust varies as load times span squared divided by eight times the rise: halve the rise and you double the sideways push for the same load. A jack arch is that relationship taken to its limit, which is why a nearly flat arch is the most demanding thing on this page and not, as it looks, the simplest.

So the arch is never the whole job. What stands beside it matters as much as what spans the opening. A pier between two windows carrying an arch on each side has the two thrusts pushing against one another and largely cancelling; the arch at the end of a run has nothing on the far side and drives its thrust into a jamb that may be a single brick wide and stop at a corner a metre away. Openings near a return, a movement joint or the end of a wall are the ones that come back. Masonry arch analysis puts it as a line of thrust: the arch stands as long as a thrust line can be traced that lies entirely inside the depth of the ring, a position set out in Jacques Heyman's The Stone Skeleton, and keeping that line inside the middle third of the ring depth is what avoids tension at the joints.

Because of that, arches fail by hinging rather than by crushing. Four hinges anywhere along the ring convert it from a structure into a mechanism, and the movement needed to start the process is very small — a few millimetres of outward spread at an abutment will do it. That is the classic picture on an old elevation: a hairline running out of one haunch, the key sitting proud or low, and a crack through the spandrel above. Nothing crushed; the geometry simply stopped being able to hold a thrust line.

Where the arch has to carry a floor, a roof, or brickwork much above its own height, this stops being a bricklayer's decision and becomes an engineer's. TMS 402/602 Building Code Requirements and Specification for Masonry Structures governs the design in the American system and BS EN 1996-1-1 Eurocode 6 in the European one, under whatever general masonry provisions the local authority has adopted. Take the ring depth, the number of rings and the abutment check from whichever applies, not from what the last arch happened to be.

What each band of rise-to-span asks for on site
Rise ÷ spanWhat it isWhat it demands
About 1/8 or lessFlat or jack archVery high horizontal thrust; skewbacks cut into substantial abutments, or a steel angle doing the real carrying, plus a built-in camber so it does not read as sagging
Between 1/8 and about 1/2Segmental archThe everyday window head; radius struck from a point below the springing line, moderate thrust, and a taper small enough that a rough ring is often workable
Exactly 1/2Semicircular archCentre of curvature on the springing line, thrust largely vertical at the springing, 180 degrees of ring to fill and the largest brick count of the three
Greater than 1/2Stilted, horseshoe or pointed workOutside the three common families; setting out has more than one centre, and the voussoirs stop being interchangeable
What each band of rise-to-span asks for on site

What Comes Apart, and What Was Never Meant to Stay

Five things sit over that opening while the work is going on, and only three of them are still there at the end. The ring of voussoirs and the brickwork it carries are permanent. The centre, the bearers under it and the folding wedges under those are temporary works — they exist to hold the geometry until the mortar can, and in the United Kingdom their design and their removal fall under BS 5975 Code of Practice for Temporary Works Procedures and the Permissible Stress Design of Falsework rather than under anybody's judgement on the day.

That split is also the split in the take-off. The permanent items are counted in bricks and in mortar; the temporary items are counted in sheet material, timber and hire, and they are the ones routinely left off a price because they do not show up in the finished elevation.

The parts of an arch over a window head

A segmental arch over a window opening, taken apart into five parts: the brickwork carried above the extrados, the ring of voussoirs, the timber centre struck to the rise, the bearers and folding wedges beneath it, and the jambs the whole assembly pushes against.
  1. Brickwork over the extrados — the load the arch was built to carry, and the reason the ring must not be eased until this work is up and the mortar in it has stiffened Brick Calculator
  2. Ring of voussoirs and key — counted by the angle each brick sweeps rather than by length, and cut to a taper set by the ring depth and the radius Radial Arch Brick Wedge Taper Calculator
  3. Centre or turning piece — cut to the intrados line and therefore to the rise; any sag in it during the turn is permanently built into the finished soffit Brick Arch Rise-to-Span Ratio Calculator
  4. Bearers and folding wedges — the only adjustable part of the assembly, and the reason the centre can be lowered evenly from both ends instead of knocked out
  5. Jambs and skewbacks — where the inclined thrust lands; the arch is only ever as good as the masonry standing either side of the opening Masonry Pier Axial Capacity Calculator

Striking the Curve on a Sheet of Ply

Two lines of geometry convert the two numbers you have into the two numbers the rest of the job needs. The radius to the soffit is the span squared plus four times the rise squared, all divided by eight times the rise. Half the included angle is the arcsine of half the span divided by that radius; double it for the total sweep. A 1.4 m opening with a 260 mm rise gives a radius of about 1.07 m and a total sweep of about 82 degrees — which puts the centre of curvature more than a metre below the springing line, out on the scaffold boards or in mid-air rather than anywhere you can put a nail.

That is the practical problem with striking a segmental curve, and there are two honest answers. If the centre point does land on solid deck, a trammel — a batten with a pin at one end and a pencil at the other — draws the line directly and is the most accurate method available on site. If it does not, set the curve out by offsets instead: mark the chord across the ply, then step along it and measure the perpendicular distance from the chord up to the curve at each station. The offset at the middle of any chord on a circle of known radius is a fixed quantity, so the same arithmetic that draws the curve also checks it afterwards.

Build the centre stiff. For a narrow opening a single turning piece cut from ply or solid timber is enough; wider spans want two ribs with a spacing of laggings across them, and thicker material than looks necessary, because the ring you are about to build on it is heavy and any deflection it causes is permanently in the soffit. Set the centre on bearers carried by the jambs or by props, with folding wedges — two opposed tapered pieces — between bearers and centre at each end. Level the centre off the springing line at both ends, not off the floor, and check it across the width as well as along the span.

Then check the struck line before a single brick goes on it. Lay a straightedge as a chord between two marked stations and measure back to the curve at the midpoint of that chord; do it at three or four positions across the arc. A curve that is right at the springings and wrong in the haunches will look convincing on the bench and impossible on the elevation, and every subsequent voussoir inherits the error.

  1. Measure the finished span between the jambs at the springing line, not at the reveals.
  2. Agree the rise, then work the radius and the total included angle from it before touching the ply.
  3. Strike the intrados on the ply with a trammel if the centre point is reachable, or by chord offsets if it is not.
  4. Check the struck line by chord and mid-ordinate at three or four stations before cutting.
  5. Cut the centre, brace it across its width, and set it on bearers and folding wedges levelled to the springing line at both ends.
  6. Mark the voussoir divisions on the face of the centre itself so the ring can be laid to marks rather than to eye.

The chord between two check stations and the offset from that chord to the curve are the two measurements a struck arc can be proved with — useful for drawing the line, and the only way to audit it once the trammel is put away.

The radius the arc is supposed to have been struck at.

The distance along the arc from one check point to the next.

How much of the curve you are checking, measured around it.

Mid-ordinate at each station

0.8348 in

High confidence

A chord reading short against these figures means the pivot moved or the trammel shortened; a mid-ordinate reading high means the radius struck was tighter than the design. The two together tell you which.

Chord between adjacent stations
3.5 ft
Check stations that fall on the arc
11 stations
Angle the arc turns through
101.57 degrees
Chord across the whole arc
34.09 ft

Add the equipment this sizes

This result is a specification — 0.8348 in — not a quantity. Put the thing it sizes into your project: how many, what you call it, and your supplier’s price.

44 ft
Schematic, drawn to the proportions you entered — not to scale on screen.

What this calculation does not cover

  • Arcs longer than a semicircle are reported at the clamp, where the chord equals the diameter — check such a curve in halves instead.
  • This proves the curve against its own radius. Whether the pivot was in the right place is a separate check, made from the goal or the datum rather than from the arc.

Dividing the Curve Into Bricks

A ring is not divided by length; it is divided by angle. Each brick occupies an arc at the soffit equal to its own face dimension plus one soffit joint, and the angle that arc subtends is simply that length divided by the radius, in radians. Divide the total included angle by the angle per brick and you have the count for one ring. Feed the arithmetic the face plus the joint rather than the bare brick, or the count comes out optimistic and the ring closes short.

Adjust the answer to an odd number. An odd count puts one brick centred on the crown as a key, which is both the traditional appearance and a genuinely useful thing during the turn, because the two halves of the ring are then mirror images and can be built together. Take the adjustment out of the joint width rather than out of the bricks: a millimetre or two spread across two dozen soffit joints is invisible, whereas a thin slip beside the key announces itself from the far side of the street for the life of the building.

The taper follows from the same two numbers and one more. Because the back of the ring travels on a larger circle than the soffit, each brick has to be wider at the back than at the front, and the difference is the ring depth multiplied by the arc each brick occupies, divided by the radius. A half-brick ring 102.5 mm deep on a 2 m radius, with 65 mm plus a joint to the curve, tapers about three and a half millimetres — comfortably absorbed by opening the joint from six millimetres at the soffit to under ten at the back. Turn the ring on edge so it is 215 mm deep on a 1 m radius and the same brick now tapers about fifteen millimetres, which is a different proposition entirely.

That is where the rough arch stops working. A ring built from uncut bricks relies on the mortar joint alone to absorb the taper, and there are limits at both ends: a joint too thin at the soffit has no bed left in it, and a joint too wide at the back is a shrinkage crack waiting to happen and looks like one long before it becomes one. Brick Industry Association Technical Notes on Brick Construction 31: Brick Masonry Arches sets out the accepted minimum at the soffit and maximum at the back — around an eighth of an inch and three-quarters of an inch are the figures usually quoted — and those two limits, not preference, are what decides whether the bricks get cut.

Multi-ring work multiplies the problem rather than sharing it. Each ring stands on its own radius, so each has its own angle per brick, its own count and its own taper, and the outer ring of a two-ring arch is a longer arc with a shallower taper than the inner one. Work them separately and never assume the outer count is the inner count plus a couple. The same applies to the second leaf of a solid wall: an arch turned in both leaves is two arches at two radii that happen to be built at the same time.

Hand it the soffit radius and the total sweep worked out in the previous section, along with the brick's arc face and the ring depth, and it returns the count and the taper per brick — round the count to odd yourself, and treat the taper as the number that decides cut or uncut.

The radius to the inner (intrados) face of the arch.

The brick's length as measured along the curve.

The brick's depth in the radial direction (arch thickness per ring).

The total angle the arch spans, in degrees.

Bricks needed for the arch

32 bricks (per ring)

Medium confidence

Assumes a constant mortar joint width along the inner face and a wedge-cut (or tapered joint) to close the gap at the outer face — for a very tight radius, verify the taper doesn't exceed what a simple tapered mortar joint can accommodate without actually cutting wedge-shaped bricks.

Angle per brick
5.69 °
Wedge taper per brick
0.4 in

Estimated cost — your price

This site holds no price list for this material — local prices vary too much to publish honestly. Enter your supplier's price and the result is costed with it.

180°6.5 ft1.98 m

What this calculation does not cover

  • The calculation carries no mortar joint. The brick length you enter is taken as the whole spacing along the inner face, so enter brick plus bed joint if you want the count and taper to describe a jointed arch - enter the bare brick and you get the figures for bricks laid touching.
  • This is setting-out geometry, not a structural check. It says nothing about whether the ring depth is adequate, how much horizontal thrust the arch drives into the abutments, or what the centering has to carry until the mortar has gained strength.
  • The answer is one ring of bricks at the radius you enter, one brick deep through the wall. A second or third ring sits at its own larger radius and needs its own run, and an arch through a wall thicker than one brick needs the count multiplied for that extra depth.
  • Bricks are rounded up to the next whole one and nothing is added for cutting waste or breakage. The leftover fraction is not set out for you: no keystone is allowed for, the count is not forced odd, and on site the odd part-brick is normally absorbed by adjusting joint widths around the arch.
  • It covers segmental and semicircular arches - span angles from 30 to 180 degrees and radii up to 10 m (33 ft). Flat and jack arches fall outside that, and a figure typed beyond a bound is pulled back to the bound before the answer is worked out. Each brick is also treated as an arc rather than the straight chord it really is, which puts the count slightly high on tight radii.

Cutting the Wedges

One fact makes this much easier than it looks: on a circular arch every voussoir is the same wedge. The taper does not vary from springing to crown, so a single template serves the whole ring, and the only bricks that differ are the two at the springings, which are cut to sit square on the skewbacks. Elliptical, three-centred and Tudor arches do not share that property — their radius changes along the arc — and a template made at the crown of one of those will be wrong by the time it reaches the haunch. Establish which kind you have before making a template, because it is the difference between one pattern and a dozen.

Cut wet on a bench saw with the template clamped or scribed, keep the cut square through the depth of the brick so the bed is not wound, and lay the ring out dry on the boards in the order it will be built. Traditional axed work does the same job with a bolster and a scutch and accepts a coarser fit; gauged work is a different trade again, using soft rubbers rubbed to shape on a stone and bedded in lime putty joints of a millimetre or two, and it is worth knowing that a gauged arch is repairable in a way that a hard-mortar cut arch is not. Historic England's Practical Building Conservation: Earth, Brick and Terracotta is the reference for that end of the work and for matching existing arches on repair.

Purpose-made arch bricks are the third route and the one that has to be decided earliest. In the British system BS 4729 Clay and Calcium Silicate Bricks of Special Shapes and Sizes — Specification covers the standard specials, with anything outside it made to drawing; either way the lead time comes off the manufacturer's own production schedule and is measured in weeks, not days. Order the ring bricks separately from the general facings whichever route you take, and add breakage on top of the count — a cut wedge that chips at the soffit arris is scrap, because that arris is the visible edge on the finished soffit.

Turning It

The ring goes up from both springings at once, alternately, never one side and then the other. Build the skewbacks first — the inclined beds cut into the jambs that the first voussoir sits on — and check that their faces are square to the ring, because a skewback out of angle puts the first brick out and every brick after it inherits the error with interest. Bed each voussoir on a full joint worked from the back so it is solid to the soffit; mortar pushed in from the face afterwards leaves a void behind a neat-looking line, and a void in an arch ring is the place a hinge will find first.

Keep the face plane as ruthlessly as the curve. A voussoir tapped in with the trowel handle stays where it is put; one persuaded with the blade rocks on its bed and drifts. Check the face against a straightedge across the ring every few bricks and check the soffit against the centre with a light behind it — a brick standing proud of the intrados by two millimetres is visible on a rendered reveal and impossible to correct once the key is in.

  1. Cut and check the skewbacks, then set the centre and wedge it dead to the springing line.
  2. Mark the voussoir divisions on both faces of the centre and dry-lay the ring to confirm the count closes on a key.
  3. Build alternately from both springings toward the crown, bedding full joints worked from the back.
  4. Fit the key last, driven home only enough to bed it — a key hammered tight lifts the haunches off the centre.
  5. Build the spandrel and the brickwork above the arch while the centre is still carrying it.
  6. Leave the centre in place until the mortar has the strength the specification requires, then ease the wedges.

Easing the Wedges

Striking the centre is the moment the arch becomes a structure, and it is a scheduled operation rather than a tidying-up job at the end of the week. The arch has to take its own weight and the weight of whatever stands above it in one movement, and mortar that has not developed enough strength will simply squeeze at the soffit joints and let the ring drop. The timing comes from the mortar and the specification — ASTM C270 Standard Specification for Mortar for Unit Masonry describes the material but not the striking time, and lime mortars need very much longer than a cement-lime mix. Nothing about it is judged by looking at the joint.

Ease, do not knock. Folding wedges exist so the centre can be lowered a millimetre at a time, from both ends alternately, keeping it level as it comes away. Dropping one end first loads half the ring, which is the same four-hinge geometry as an abutment spreading, only faster. Some settlement is expected and correct: the joints close up slightly, the ring beds down onto its skewbacks and the arch finds its thrust line. What you are watching for is anything asymmetric — one haunch closing and the other opening, or the key moving relative to its neighbours.

Then leave it alone. The centre comes out sideways once the wedges are slack, and the opening should not be loaded, propped against or used to carry a scaffold tie for as long as the specification says. Cover the top of unfinished work over the arch when you leave: rain running down into an open ring washes fines out of joints that are still green, and on a soft facing the staining that follows is on the most conspicuous course of the elevation.

The Arch That Is Carrying Nothing

A large share of the arches turned today are not structural at all. In veneer construction the opening is spanned by a steel angle and the ring is built off it, so the bricks are cut to a taper for appearance while the angle does the carrying. That does not make the geometry optional — the ring still has to close on a key and still has to be struck from a real radius — but it moves the failure mode. What fails on a supported arch is the support.

Steel carrying masonry is held to much tighter deflection limits than general framing, because brickwork cracks long before a beam is uncomfortable to walk on; the limits usually applied are a fraction of span in the region of one six-hundredth together with a small absolute cap, and Brick Industry Association Technical Notes on Brick Construction 31B: Structural Steel Lintels is where to take them from rather than from a joist table. Bearing length at each end matters as much as the section, and the angle's horizontal leg has to support the ring without its toe showing under the soffit.

The other veneer detail that gets missed is water. Flashing runs over the arch rather than through it, stepping up and across the ring and turning up behind the drainage plane, with weeps immediately above it at the spacing the adopted code sets. Skip it and every drop that gets through the ring — and a ring with wide back joints passes a good deal — collects on the steel, which then rusts, expands and lifts the arch off its bearing. Allow for movement too: clay brickwork grows after it leaves the kiln while steel does not, so the joint arrangement around the opening has to let the arch move relative to what carries it.

Reading One That Has Already Gone Wrong

Arches record their own history, which is a gift on a repair survey. A key sitting low with fine cracks radiating from the haunches is a spread abutment — look at what is beside the opening and what changed there, not at the arch. Wide open joints at the back of the ring with tight ones at the soffit mean the ring was turned as a rough arch beyond the taper a joint can carry. A soffit that dips in the middle while the joints stay closed is a centre that deflected during the turn and was never going to recover.

Two more are worth naming because both get misdiagnosed as failure. Staining and efflorescence concentrated on the arch alone usually mean water standing on the back of the ring, which is a flashing problem above rather than a masonry problem in front of it. And a jack arch that appears to sag when a straightedge says it is dead level is doing exactly what it should — jack arches are built with a slight upward camber for that reason, proportional to the span, taken from the arch supplier's shop drawing or from Technical Notes 31 rather than from the bricklayer's eye.

Pricing an arch before the rise is settled

Everything on this list moves when the rise moves, which is why an arch quoted from a span alone is a guess. Fix the ratio first, then let the ring, the cutting, the centring and the abutment check fall out of it.

  • Voussoirs per ring, counted by angle and rounded to odd — Arc consumed per brick is the face plus one soffit joint. An even count means no key; take the correction out of the joints, never out of a brick.
  • Cutting route decided from the taper, not from preference — Ring depth times arc per brick divided by radius. Under the accepted back-joint limit a rough ring works; over it, you are cutting wedges or ordering specials with a lead time.
  • Second and subsequent rings priced separately — Each ring sits on its own radius, so it has its own count and its own taper. Two rings is not twice the first ring.
  • Centring, bearers, folding wedges and props — Sheet material, timber and hire — temporary works that never appear in the finished elevation and are the line most often left out of the price.
  • Abutment check and any lintel or angle behind the face — Flat and jack arches drive most of their load sideways. What stands beside the opening, and whether steel is carrying the ring, belongs in the price and in the programme.
  • Breakage on cut voussoirs, over and above waste — A chipped soffit arris is scrap because that edge is visible on the finished soffit. Order ring bricks separately from the general facings.
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Opens the calculators above on one screen with the dimensions from this article already filled in. Quantities only — this site publishes no price list, because local prices vary too much to publish honestly.

Drawn from

  • Brick Industry Association, Technical Notes on Brick Construction 31: Brick Masonry Arches
  • Brick Industry Association, Technical Notes on Brick Construction 31B: Structural Steel Lintels
  • TMS 402/602 Building Code Requirements and Specification for Masonry Structures
  • International Residential Code, Section R606 General Masonry Construction (as adopted and amended locally)
  • BS EN 1996-1-1 Eurocode 6: Design of Masonry Structures — General Rules for Reinforced and Unreinforced Masonry Structures
  • BS 4729 Clay and Calcium Silicate Bricks of Special Shapes and Sizes — Specification
  • BS 5975 Code of Practice for Temporary Works Procedures and the Permissible Stress Design of Falsework
  • ASTM C270 Standard Specification for Mortar for Unit Masonry
  • Jacques Heyman, The Stone Skeleton: Structural Engineering of Masonry Architecture
  • Historic England, Practical Building Conservation: Earth, Brick and Terracotta
  • OSHA 29 CFR 1926 Subpart Q Concrete and Masonry Construction

Guidance, not a specification. Local codes, the engineer of record and the product manufacturer’s instructions govern where they differ from anything written here.