A circular arch is described by three numbers, and only two of them are ever independent. Give the span and the rise and the radius follows. Give the span and the radius and the rise follows. The arch calculator above solves whichever one you are missing, then produces the things that actually get used on site: the arc length, the included angle, the area of the opening below the curve, and a set of ordinates for marking the shape out full size.
Arb Digital publishes this as a setting-out and teaching tool. It is geometry. An arch is a structural element that carries load by compression and pushes outward at its springings, and this page computes none of that: it does not calculate thrust, it does not size an abutment, a tie or a ring, and it does not tell you whether any arch will stand. Structural design of an arch, in masonry or anything else, belongs to a licensed engineer working to the code adopted where the building is.
What This Arch Calculator Does
It works with the segmental case, which is the general one: a circular arc struck between two springings, with the crown some rise above them. A semicircular arch is the special case where the rise equals half the span, and the tool reports it as such when the numbers land there. A very shallow jack or camber arch is the other extreme, and the same equations hold.
The outputs split into two groups. The first is the description of the curve — radius, included angle, arc length, segment area. The second is the setting-out data: heights of the curve above the springing line at equal steps across the span, and the even division of the arc into ribs or voussoirs. Those ordinates are the practical heart of the page, because on most real arches the centre of the circle is somewhere below the floor slab, and swinging a radius from it is not possible.
This is not the same job as the abstract circle tools we already publish. The circle calculator converts between radius, diameter, area and circumference. The arc length calculator and sector area calculator work from a radius and a central angle, and the chord calculator from a radius and a chord. None of them take a span and a rise, and none of them produce marking-out ordinates. That is the boundary: those pages are geometry in the abstract, this one is geometry expressed the way a builder's drawing states it.
How to Use It
- Pick what you are missing. Most drawings dimension the opening and the rise, so radius is the usual unknown.
- Measure both values from the springing line. The rise is the height of the crown above the springing, not above the floor and not above the head of the opening.
- Read the included angle before you go further. It tells you at a glance whether the arch is shallow, segmental or semicircular, and it is what the arc divides into for voussoirs.
- Take the ordinates for the marking-out. Snap the springing line on a sheet, step off the equal divisions, square up each ordinate and bend a batten through the points.
- Send the loads to somebody qualified. The shape is a geometry problem; the thrust, the abutment and the ring are not.
The Formula / How It's Calculated
With span S and rise h, the radius of the circle through the two springings and the crown is r = (S²/4 + h²) / (2h). Going the other way, with a known radius, the rise is h = r − √(r² − S²/4), which requires the radius to be at least half the span, and the span is S = 2√(h(2r − h)).
The included angle follows from the half-chord: θ = 2 × arcsin(S / 2r) for a rise up to half the span. Once the rise exceeds half the span the arc is the major one and the angle becomes 2π − 2 arcsin(S / 2r), which the tool handles. Arc length is then rθ with θ in radians, and the area of the segment between the chord and the arc is r²(θ − sin θ) / 2. Those two segment relationships are set out at Wolfram MathWorld's Circular Segment entry.
The ordinates come from the circle itself. Measuring x from the centre of the span, the height of the curve above the springing line is y = √(r² − x²) − (r − h). At the middle that returns the rise; at the springings it returns zero.
Worked example with the loaded values. A span of 3000 with a rise of 750 gives r = (2,250,000 + 562,500) / 1500 = 1875. The included angle is 2 arcsin(1500 / 1875) = 2 arcsin(0.8) = 106.26 degrees, or 1.8546 radians, so the arc length is 1875 × 1.8546 = 3477.4. The segment area is 1875² × (1.8546 − 0.96) / 2 = 1,572,522. The springing angle, the tilt of the arch face where it meets the wall, is half the included angle at 53.13 degrees from horizontal.
Why Ordinates Beat Swinging a Radius
The classical way to draw an arch is to find the centre and swing it with a trammel. For a semicircular arch over a doorway that is easy: the centre sits on the springing line, in mid-air but reachable. For a shallow segmental arch it is not. A 3000 opening with a 300 rise has a radius near 3900, and the centre is 3600 below the springing line — through the slab, into the ground, usually outside the building.
Ordinates sidestep the whole problem. Snap the springing line and the centreline on a sheet of ply, divide the half-span into equal steps, mark the calculated height at each, and spring a thin lath through the marks. Eight divisions is enough for a shallow curve; sixteen for anything approaching a semicircle, where the ends turn sharply and a lath through widely spaced points will cut the corner. The tool caps the divisions at twenty-four because beyond that the marking error exceeds the geometric gain.
The other reason to work in ordinates is that they are self-checking. The middle ordinate must equal the rise and the end ordinates must be zero. If they are not, something upstream is wrong, and you find out at the drawing board rather than after the formwork is cut. For the plywood take-off, our plywood sheet calculator covers the sheet count.
Voussoirs, Ribs and the Even Division of an Arc
Dividing the arc evenly is the second half of arch setting-out. In masonry the divisions are voussoirs, the wedge-shaped units of the ring, and there is a long-standing preference for an odd number so that a single keystone lands on the centreline rather than a joint. In formwork the divisions are ribs or plywood segments, and in a glazed or panelled arch they are the panel joints.
Two numbers matter. Each division subtends the included angle divided by the count, and each occupies the arc length divided by the count. For a brick ring, the joint at the intrados is narrower than the joint at the extrados by the ratio of the two radii, which is why a masonry arch built with parallel-sided units and no taper opens at the back or crushes at the front. The tool reports the per-division angle and arc so the taper can be set out from it.
Where the arch is a fabricated rib rather than masonry, the same division sets the segment chords, and the difference between a chord and its arc — the versine of the small segment — is what decides how many facets are needed before a curve reads as smooth. That versine falls with the square of the division length, so doubling the number of segments cuts the visible flat by roughly a factor of four.
What This Page Deliberately Does Not Do
An arch is a compression structure. It carries vertical load down to its springings, and in doing so pushes horizontally outward against whatever is there. That thrust is real, it grows sharply as the arch gets shallower, and it is what pushes walls out and cracks abutments when it is not resisted. A tie rod, a buttress, a mass of masonry or an adjacent bay of structure has to take it.
None of that is calculated here, and none of it should be inferred from the geometry. This page will not tell you the thrust, the ring thickness, the depth of an abutment, the size of a tie, the load a centering must carry while the mortar cures, or whether an existing arch is stable. Those are structural design questions governed by the building code adopted locally — in much of the United States the family of codes published through the ICC Digital Codes library, with loads from ASCE/SEI 7, Minimum Design Loads and Associated Criteria for Buildings and Other Structures — and amendments differ by state, province and country. A licensed engineer decides them, and existing masonry arches in particular need someone on site rather than a formula. For related preliminary work our beam load calculator and retaining wall calculator carry the same framing and the same limits.
Arb Digital builds free calculators and interactive tools that earn organic search traffic for construction, masonry and joinery businesses. Browse the library, or tell us what your customers keep asking you to work out.
Browse Free Tools Talk To Arb DigitalCommon Mistakes to Avoid
- Measuring the rise from the floor — it is the height of the crown above the springing line, and using the floor inflates the radius badly.
- Entering a radius smaller than half the span — no circular arc can span further than its own diameter, and the tool says so rather than returning a meaningless number.
- Using too few ordinates near a semicircle — the curve turns fastest at the springings, so a lath through widely spaced points cuts the ends flat.
- Dividing the ring into an even number of voussoirs — it puts a joint on the centreline instead of a keystone, which most masonry practice avoids.
- Treating the geometry as a design — an arch thrusts outward, and nothing on this page tells you how much or what resists it.
Related Free Tools From Arb Digital
For the abstract circle relationships behind this page, use the circle calculator, the arc length calculator, the sector area calculator and the chord calculator. For the rest of the opening, the roof pitch calculator and concrete block calculator help with the surrounding work, and the angle cut calculator handles the angled cuts on the centering. The full list is on the free online tools hub.
Frequently Asked Questions
Use r = (S squared divided by four, plus h squared) divided by twice h, where S is the span and h is the rise above the springing line. For a semicircular arch that reduces to exactly half the span.
The level at which the curve of the arch begins, where it meets the supporting wall or pier. Span and rise are both measured from it, and heights taken from the floor instead are the most common source of error in arch setting-out.
Because a circular arc cannot span further than the diameter of its own circle. If the radius you have is less than half the span, either the radius or the span is wrong, or the shape is not a single circular arc.
They are the heights of the curve above the springing line at equal steps across the span. They let you mark an arch out full size without finding the centre of the circle, which on a shallow arch is usually below the floor and outside the building.
That is a masonry design decision, not a geometric one. What the geometry gives you is the angle and the arc length each division occupies, and the convention of using an odd count so a keystone rather than a joint lands on the centreline.
No. An arch pushes outward at its springings, and that thrust increases as the arch becomes shallower. Calculating it and resisting it with abutments or ties is structural design work for a licensed engineer.
No. Geometry says nothing about the condition of the masonry, the mortar, the abutments or the foundations. An existing arch that concerns you needs a structural engineer on site.
Those start from a radius and a central angle and work in the abstract. This page starts from the span and rise that a building drawing actually gives, solves for the missing quantity, and adds setting-out ordinates and ring divisions.
This tool computes the geometry of a circular arch from dimensions you supply, for setting-out and education only. It is not a structural design, it calculates no thrust, load or capacity, and the structural design of any arch must be carried out by a licensed engineer in accordance with the building code adopted locally.
