Arch Calculator

Every true segmental arch — the shallow curve over a window, a doorway, a culvert — is a slice of one circle, and two measurements describe it completely: the span it crosses and the rise at its crown. This calculator recovers the rest of the circle from those two: the radius your formwork must be built to, the arc length the material must follow, the spring line angle, and the chord — for masons, carpenters, steelworkers and anyone laying out a curve that has to meet its other half.

One Circle, Two Measurements

A segmental arch is a circular arc of less than a semicircle, and circle geometry is miserly with its secrets: specify the chord (your span) and the sagitta — the rise, the arrow-height of the arc — and the radius is determined by the intersecting-chords theorem, in the form builders have used for centuries: R = (a² + h²) ÷ (2h), where a is the half-span and h the rise. From the radius, everything else on the arch derives: the included angle (2·asin(a/R)), the arc length (R × angle), and the centre's position — which sits R − h below the spring line, the fact that makes full-size layout possible with a nail, a string and a pencil. The calculator's three outputs are therefore not three calculations but one: the circle, confessed by its two most measurable features.

Worked Example: A Ten-Foot Opening, Two and a Half Up

The defaults describe a handsome garden-wall or window arch: span 10 ft, rise 2.5 ft (a quarter of the span, the proportion classical segmental arches favour). Half-span a = 5; radius = (25 + 6.25) ÷ 5 = 6.25 ft; the centre sits 6.25 − 2.5 = 3.75 ft below the spring line; the included angle is 2 × asin(5/6.25) = 2 × 53.13° ≈ 106.3°; and the arc length is 6.25 × 1.855 = 11.59 ft — the length of brickwork, steel or timber that must follow the curve, against the 10 ft of straight opening it closes. Note what the radius teaches: a shallow arch belongs to a surprisingly small circle (a 2.5 ft rise on 10 ft of span still turns through more than a quarter of its generating circle's right half), which is why eyeballing the curve fails and the struck layout — string from the buried centre — succeeds.

Laying the Arch Out Full-Size

The outputs are a layout recipe. Find the centre: on your formwork or floor, mark the spring line, drop the centreline perpendicular, and measure R − h down from the spring line along it (3.75 ft in the example) — that point is the pivot. A trammel bar, a batten with two nails, or the mason's string of length R struck from the pivot draws the intrados (the arch's underside curve) exactly; the extrados (top curve) is the same swing at R plus the arch's depth. Where the centre falls inconveniently far below (very flat arches bury it metres under the work — a 1 ft rise on a 12 ft span puts the pivot 18 ft down), the trade switches to ordinate layout: compute the curve's height above the spring line at intervals along the span from the circle's equation — points this calculator's geometry supplies — and fair a batten through them. Both methods are the same circle; the calculator's radius is the licence for either.

Rise Ratios and the Arch's Manners

The rise-to-span ratio is the arch's character and its structural temperament in one figure. Semicircular (rise = half the span, ratio 1:2) is the Roman manner — vertical springing, loads delivered kindly downward. Segmental arches at 1:4 to 1:6 (the worked example's family) flatten the profile, span wider openings under lower headroom, and pay for it in thrust: the flatter the arch, the more its load resolves sideways into the abutments, which is why flat arches live in thick walls, between buttresses, or behind tie rods. Below about 1:8 the profile becomes a jack arch — nearly flat, wholly dependent on its abutments and often on hidden steel. The calculator stays neutral among the manners (it prices any rise you enter), but the ratio is the number to say out loud before building: geometry will draw any arch you specify, while physics grades the abutments you gave it.

From Arc Length to a Brick Count

The arc becomes an order in one more division. A brick arch's voussoirs are counted along the intrados: divide the arc length by the brick-plus-joint module measured on the curve. For the worked example — arc 11.59 ft = 139.1 in, standard brick-and-joint module of 8.375 in along the curve — the count is 139.1 ÷ 8.375 ≈ 16.6, which the tradition rounds to the next odd number, 17 voussoirs, so a single key brick sits centred at the crown with the joints mirroring down both haunches. The extrados tells the complementary story (the same bricks splay wider at the top: at an arch depth of 8 in the outer arc runs about 12.8 ft), which is why tapered voussoirs and wedge-shaped joints exist. Stone and precast orders run the identical division with their own modules — the arc length is the input the whole materials chain consumes.

Thrust, Estimated in One Line

For a feel of the forces the abutments must swallow, the parabolic approximation serves: horizontal thrust H ≈ wL² ÷ (8h) for a uniformly loaded arch of span L, rise h, carrying w per foot. The worked example under a hefty 500 lb per foot of load: H ≈ 500 × 100 ÷ 20 = 2,500 lb per foot pushing outward at each springing — and the formula's moral is its denominator: halve the rise and the thrust doubles, which is the flat arch's bargain stated in pounds. Real thrust depends on the arch's exact curve, fill and load pattern (engineers use more exacting methods), but the estimate sizes buttresses honestly enough to explain why they exist.

Related Tools

More from the Construction Calculators hub: the Bolt Circle Calculator — the same circle, drilled instead of built — and the Square Feet to Cubic Yards Calculator for the concrete the arch may carry.

Frequently Asked Questions

How do I find an arch's radius from span and rise?

With the formula this page runs: halve the span (a), then R = (a² + rise²) / (2 × rise). For the worked example: (25 + 6.25)/5 = 6.25 ft. The derivation is the intersecting chords of the generating circle — the sagitta relation h(2R − h) = a² rearranged — and it inverts just as cleanly: given a radius and a span, the rise is R − √(R² − a²), the form to use when matching new work to an existing curve measured by its chord.

What rise should a segmental arch have?

Between a quarter and a sixth of the span is the classical working range for load-bearing segmental arches — enough curve to behave like an arch, flat enough for useful headroom. Decorative and veneer arches cheat lower (they carry only themselves, often on lintels doing the real work), while structural spans under serious load rise toward the semicircle. Enter your constraints honestly: the headroom you need sets the ceiling for rise; the abutments you have set the floor.

How long are the bricks or boards along the curve?

That is the arc length output — 11.59 ft in the example — and it is the quantity the straight span conceals: materials follow the curve, so ordering, cutting lists and voussoir counts all key off the arc. Brick arches divide the arc by the brick-plus-joint module to count voussoirs (typically an odd number, so a key sits at the crown); timber and steel order to the arc plus working allowance.

Can this calculator do a semicircular arch?

It identifies one: enter a rise equal to half the span and the formula returns R equal to the half-span with an included angle of 180° — the calculator's note names it. Gothic (pointed) and elliptical arches are different geometries (two centres, or a true ellipse) and are not this page's circle; a segmental calculator asked to price them would be lying with great precision, which this hub avoids on principle.

Where exactly is the centre for a flat arch?

Distance R − rise below the spring line, on the centreline — for a 12 ft span at 1 ft rise, R = (36 + 1)/2 = 18.5 ft, so the pivot lies 17.5 ft below the work: inside the building, under the floor, in the neighbour's garden. That inaccessibility is precisely why flat arches are laid out by ordinates (computed offsets from the spring line) rather than struck with a string, and why their geometry rewards a calculator over a tape measure.

Does the arch's thickness change the radius I need?

There are two radii — intrados (underside, this calculator's, the one the opening sees) and extrados (top, equal to intrados radius plus the arch's radial depth). Formwork is built to the intrados; material volumes and the outer profile use the extrados; both swing from the same centre the calculator locates. State which radius a drawing means before cutting to it — the two differ by exactly the arch depth, and confusing them is a traditional way to build an arch that fits its opening only in the drawings.