Arch Calculator
Enter the span and rise (or span and radius) for any arch type and this calculator returns the radius, arc length, enclosed area, central angle, and, for elliptical arches, the focus points needed to draw the curve with a string. Switch between metric (mm) and imperial (inches) at any time.
Formula
Worked example
A segmental arch with span 1200 mm and rise 400 mm: r = (1200^2/4 + 400^2) / (2 x 400) = (360000 + 160000) / 800 = 650 mm. Half-angle = asin(600/650) = 67.38 deg, central angle = 134.75 deg. Arc length = 650 x (134.75 x pi/180) = 1528 mm. Enclosed area = 650^2/2 x (2.352 - sin(134.75 deg)) = 422500/2 x (2.352 - 0.710) = 346900 mm^2.
What is an arch calculator used for?
An arch calculator works out the geometric properties of a curved opening from two known measurements. Builders, carpenters, and bricklayers use it to find the radius needed to set a compass or trammel when cutting formwork, to calculate arc length when ordering curved brick, stone, or timber, and to check whether a rise-to-span ratio meets structural guidelines. Architects use it to convert between span, rise, radius, and arc length when coordinating drawings. The same geometry applies to window heads, door openings, bridge soffits, tunnel portals, and decorative corbelling.
Arch types and how to choose
A semicircular arch is the simplest: the rise equals exactly half the span, so the radius equals half the span too. It is structurally well-balanced and the easiest to set out. A segmental arch uses a circular arc with a rise less than the radius, giving a flatter profile. The smaller the rise relative to the span, the longer the radius and the greater the horizontal thrust on the abutments. A rise-to-span ratio below 1:8 means very high thrust and demands careful structural assessment. An elliptical arch gives a smooth oval curve. The focus-point method lets you draw it with a loop of string pinned at the two foci - this calculator gives you the focus distance. A gothic (pointed) arch uses two circular arcs meeting at a point at the apex, one springing from each side. Because thrust is directed more vertically, gothic arches transmit less outward force than semicircular ones of the same span, which is why they were favoured in medieval cathedral construction alongside flying buttresses.
How to use the arch calculator
Select your arch type, then choose whether you know the span plus the rise, or the span plus the radius (segmental arches only). Enter your measurements in millimetres (metric) or inches (imperial) and the results update instantly. For elliptical arches the calculator shows the focus-point distance so you can draw the arch with a string: push two pins into the work surface at F1 and F2 (each at the focal distance from the midpoint of the span), loop a string of total length equal to span + 2 x rise around both pins, keep the string taut with a pencil, and trace the full semi-ellipse. The optional arch depth field adds volume to the output, useful for estimating concrete, mortar, or masonry volumes. The chart shows how the radius changes as the rise varies for a fixed span, helping you choose the best profile for your project.
Setting out an arch on site
For circular arches (semicircular, segmental, gothic) the key value is the radius. Mark the centre point on the spring line: for a semicircular arch it is at the midpoint of the span; for a segmental arch it is below the spring line at a distance equal to r minus the rise. Swing the compass or trammel from that centre to mark the intrados (inside face) and extrados (outside face, offset by the arch depth or brick depth). For each voussoir (wedge-shaped unit), the angle per segment equals the central angle divided by the number of voussoirs; each radiating joint bisects that angle. Add a keystone at the apex for odd unit counts. For timber formwork, the same centre-and-radius method works on the plywood or board before cutting with a jigsaw.
Arch type comparison
| Arch type | Rise : span ratio | Typical use | Structural behaviour |
|---|---|---|---|
| Semicircular | 1 : 2 (0.50) | Doorways, windows, bridges | Well-balanced; moderate outward thrust |
| Segmental (moderate) | 1 : 4 to 1 : 2 (0.25-0.50) | Brick facades, road bridges | Lower profile; increased horizontal thrust |
| Segmental (shallow) | 1 : 8 to 1 : 4 (0.13-0.25) | Railway viaducts, flat lintels | High horizontal thrust; strong abutments required |
| Elliptical | 0.15 to 0.60 | Grand doorways, tunnels | Smooth curve; lower thrust than circular at same rise |
| Gothic (equilateral) | ~0.87 (sqrt(3)/2) | Medieval churches, cathedrals | Vertical thrust; reduced outward thrust vs. semicircular |
| Gothic (lancet/steep) | > 0.87 | Lancet windows, spires | Very vertical; minimal outward thrust |
Common arch types, their rise-to-span ratios, and typical use cases.
Frequently asked questions
What is the formula for a segmental arch radius?
The standard formula is r = (w^2/4 + h^2) / (2h), where w is the span (chord length) and h is the rise. For example, a span of 1200 mm and rise of 400 mm gives r = (1200^2/4 + 400^2) / (2 x 400) = 520000 / 800 = 650 mm.
How do I draw an elliptical arch using the string method?
Calculate the focal distance c = sqrt(a^2 - b^2) where a = span/2 and b = rise. On the spring line, mark two focus points each at distance c from the centre. Loop a piece of non-stretch string around both pins so the total loop length equals span + 2 x rise. Keep the string taut with a pencil or marking knife and sweep the full semi-ellipse. This method is accurate enough for doorways and window heads without any mathematics on site.
What rise-to-span ratio should I use?
For a semicircular arch the ratio is always 0.5 (rise = half span). Segmental arches commonly use ratios between 0.25 and 0.45 for doorways and windows. A ratio below 0.125 (1:8) produces very high horizontal thrust and requires robust abutments or tie rods. Gothic arches can exceed 0.5 and direct thrust more vertically, reducing abutment loads.
How do I calculate the arc length of an arch?
For a circular arch (semicircular, segmental, gothic lobe), arc length = radius x central angle in radians. First find the central angle: theta = 2 x asin((span/2) / radius). Then arc length = radius x theta. For a semicircular arch this simplifies to pi x radius. For an elliptical arch the exact arc length requires an elliptic integral, but the Ramanujan approximation used by this calculator is accurate to within 0.01% for typical arch proportions.
How many bricks do I need for an arch?
Divide the arc length by the length of one brick plus one mortar joint (typically 10-12 mm), rounding up to an odd number so a keystone lands at the apex. For example, an arc length of 1528 mm with 215 mm bricks and 10 mm joints gives 1528 / 225 = 6.8, round up to 7 voussoirs. If you enter the arch depth in this calculator you also get the enclosed volume, which you can use to estimate total masonry or concrete needed.
What is the difference between the intrados and extrados?
The intrados is the inner (soffit) curved surface of the arch - the curve you see from below. The extrados is the outer curved surface, separated from the intrados by the arch depth or ring thickness. Arc length in this calculator refers to the intrados. Add the arch depth to the radius to find the extrados radius if you need its arc length separately.