Brick Arch Calculator

How the brick arch calculator counts voussoirs

A brick arch is a ring of bricks set around a curve, each one pointing at the centre of the circle. Those bricks are voussoirs, and this calculator tells you how many fit around the underside of the arch (the intrados) over a given opening. It needs four numbers: the span, the rise, the size of each brick measured along the curve, and the joint between bricks. Work in feet and inches or in metres and millimetres; the maths doesn't change.

There are three steps. First comes the radius of the circle through both springing points and the crown: R = (c² + 4h²) ÷ 8h, where c is the span and h is the rise. Then the length of the curve, arc = 2R × θ, where θ = 2 × tan⁻¹(2h ÷ c) is half the angle the arch turns through. Last, the arc is divided by one brick plus one joint and rounded up to a whole brick.

Take the default imperial arch, a 4 ft opening with a 1 ft rise. The radius is (16 + 4) ÷ 8 = 2.5 ft, or 30 in. Half the angle is 2 × tan⁻¹(0.5) = 0.9273 rad, so the curve is 2 × 30 × 0.9273 = 55.64 in long. A modular brick laid on edge shows 2¼ in along the curve, and with a ⅜ in joint each brick takes up 2.625 in. 55.64 ÷ 2.625 = 21.2, which rounds up to 22 bricks: $15.40 at $0.70 each. The metric default runs the same way with a 1.2 m opening and a 0.3 m rise. R = 0.75 m, the arc is 1.391 m, and 1,391 ÷ (65 + 10) = 18.5, so 19 bricks, or 34.20 at 1.80 a brick. On the metric side the currency symbol follows your browser's region (pounds, euros, Australian or Canadian dollars) and falls back to pounds; the count and the total are right whatever the symbol.

Plenty of arch formulas write the curve as 2R × sin⁻¹(c ÷ 2R). That works until the rise passes half the span. A 4 ft opening with a 3 ft rise is a horseshoe arch, bigger than a semicircle, and the sine version quietly measures the short way round the circle: 61.2 in instead of 102.2 in, or 24 bricks when you need 39. The tangent form used here is right for shallow segmental arches, semicircles and horseshoes alike.

The brick dimension that runs along the curve

This is the input people most often get wrong. In a rowlock arch the brick lies on its long edge with the end showing; in a soldier arch it stands upright with the long face showing. Either way, the dimension running round the curve is the brick's height, not its width. For a US modular brick that is 2¼ in, as listed in the Brick Industry Association's Technical Note 10 on brick sizes. A UK standard brick is 65 mm high and an Australian standard brick 76 mm. Continental European formats vary (German bricks alone come in several heights), so measure the brick you are actually buying.

Get it wrong and the count falls well short. Enter the 102.5 mm bed width of a UK brick with a 10 mm joint and the metric default drops from 19 bricks to 13.

Joints: thin at the soffit, wide at the top

Ordinary rectangular bricks can only follow a curve if the mortar joints taper, thinner at the intrados and wider at the outer curve (the extrados). The Brick Industry Association's Technical Note 31 on brick masonry arches puts arch joints between 1/8 in (3 mm) and 3/4 in (19 mm), and suggests very uniform bricks or gauged work below ¼ in (6 mm). Because the calculator measures along the intrados, the joint you type in is the joint at the underside. ⅜ in or 10 mm is a sensible starting figure. Tighten the imperial default to a ¼ in joint and it needs 23 bricks rather than 22.

Checking the joint at the top of the ring

Do this before you commit to plain bricks. Divide the length of the extrados by the brick count, then subtract the brick height. For the metric default built as a half-brick rowlock ring, 102.5 mm deep, the outer radius is 852.5 mm and the outer curve is 1,581 mm long. 1,581 ÷ 19 − 65 leaves about 18 mm per joint at the top. At the bottom it is about 8 mm, because rounding up to 19 bricks tightens the 10 mm joint you entered. That is just inside the 19 mm ceiling.

Stand the same bricks on end as soldiers, 215 mm deep, and the top joints open to about 29 mm. Too wide. The same technical note recommends tapered bricks on short spans for exactly this reason, and adds that soldier arches are hard to fill with mortar; two or more rowlock rings are the usual fix. In imperial, the default arch in a 3⅝ in rowlock ring works out at roughly 0.28 in at the bottom and 0.58 in at the top.

Rounding, keystones and a second ring

The result is rounded up to the next whole brick, which isn't always the number you lay. Arches are normally set out with an odd number of voussoirs so that a single brick sits at the crown. The metric default already lands on 19. The imperial default gives 22, so the choice is 21 bricks with joints near 0.4 in or 23 with joints near 0.17 in. With ordinary bricks, 21 is the easier lay.

A second ring sits on a bigger radius and needs more bricks. Its inner radius is the first ring's radius plus the ring depth plus the joint between rings. On the metric default with a 102.5 mm ring and a 10 mm joint between rings, that is 862.5 mm. The curve grows in the same proportion, 1,391 × 862.5 ÷ 750 = 1,600 mm, and 1,600 ÷ 75 = 21.3, so 22 bricks at the full 10 mm joint. For an odd count, 21 is the better choice: the joints open to about 11 mm, where 23 would squeeze them to under 5 mm.

Ordering, wastage and what the count leaves out

No wastage is added. A rough arch in uncut bricks usually needs only a few spares for chipped or off-colour units, while voussoirs cut or axed on site lose more to breakage. Take the arch bricks from the same batch as the surrounding wall so the colour matches. Tapered specials made in the factory are ordered against a drawing of the arch, and the supplier confirms the count.

The figure covers the arch ring only. It leaves out the cut bricks in the spandrels either side, the centering (the temporary timber former that holds the arch up, which BIA says should stay in place for at least seven days), any lintel or steel angle behind the arch, and the flashing and weeps above it. It also doesn't check that the arch can carry the wall above. Whether an arch supports itself or sits on a lintel is a structural decision for an engineer, checked under local building rules: the IBC or IRC with TMS 402 in the US, Approved Document A in England, and the equivalent code elsewhere.

Old buildings are a case of their own. Jonathan Taylor's article on brick arches for the Building Conservation Directory explains the difference between rough arches, where the mortar does the tapering, and gauged arches built from rubbed bricks with extremely fine joints. If you are matching or repairing historic work, that difference decides which bricks you order and how many.