Work out the volume, arc length and radius of a semicircular, segmental, gothic, elliptical or flat arch. Enter span, rise and depth for a cost estimate.
| Component | Value | Unit |
|---|
Enter details and calculate to see breakdown.
Powered by Techraxy
ADVERTISEMENT
Founder & CEO, Techraxy
Faiq Ur Rahman is a web designer, digital product developer, and Founder of Techraxy, a growing platform of online calculators and utility tools. He specializes in building structured, user-friendly tools focused on finance, health, conversions, productivity, and everyday problem-solving. He is passionate about developing practical digital solutions that make complex tasks simpler for users worldwide.
User Ratings:
ADVERTISEMENT
ADVERTISEMENT
An arch is a curve with volume behind it. The face looks simple until you need a number for the radius, the arc length, or the cubic yards of masonry to fill it. This arch calculator handles five arch types, from a half-round semicircle to a flat arch, and returns the arc length, radius of curvature, central angle, face area, and total volume. Enter the span, rise, and depth, and the summary updates as you type.
Five arch types are covered. Five material types are tracked for record-keeping. Prices accept per cubic yard, per cubic foot, or per cubic meter, which covers most supplier quotes. It runs in the browser, so dimensions and prices stay on your device.
How to Use the Arch Calculator
Pick your currency from the top dropdown.
Enter the arch span, which is the total width across the opening.
Add the rise, the height of the arch above its spring line.
Set the depth or thickness of the arch ring.
Type how many identical arches you’re building.
Pick an arch type: semicircular, segmental, gothic, elliptical, or flat.
Select the material for the record, then adjust the waste factor.
Enter your price per unit volume and any additional cost.
The Summary tab leads with total volume in cubic yards and cubic meters. Detailed Breakdown lists every value the math passes through.
Each arch type uses a different geometry, but they all end the same way: face area times depth equals volume.
Formula: Semicircular radius = Span ÷ 2
Formula: Segmental radius = (Rise ÷ 2) + (Span² ÷ (8 × Rise))
Formula: Gothic radius = Span
Formula: Flat area = Span × Rise
Formula: Total volume = Face area × Depth × Count × (1 + Waste %)
The semicircle is the simplest case. Its radius is half the span, the central angle is 180 degrees, and the face area is a half-circle. Everything is exact.
The segmental arch is the one most builders actually want. Given a span and a rise, the formula solves for the radius that produces that curve. A shallow segmental arch over a wide doorway is the classic use case. The math behind it uses the chord and sagitta relationship, which is why the radius calculation looks different from the others.
The gothic arch uses a fixed geometry. The radius equals the span and each side sweeps through 60 degrees, creating the pointed peak. That’s the standard equilateral gothic profile. Real gothic arches vary, but this is the common construction shape.
The elliptical arch approximates the perimeter using Ramanujan’s formula, which is accurate to within a fraction of a percent for most ellipse shapes. Face area is π × half-span × rise, divided by two for the half-ellipse.
The flat arch is a rectangle with a level soffit. It’s not really curved at all, but it’s included because flat arches over doors and windows are common in masonry.
Material type affects display only. It doesn’t change the volume math. The material matters when you’re pricing, and the calculator lets you do that with the price per unit volume field.
Say you’re building a segmental arch over a 12 ft doorway. The rise is 3 ft. The arch ring is 1.5 ft thick. Waste factor is 10%. Concrete at $150 per cubic yard.
Span: 12 ft
Rise: 3 ft
Half span: 6 ft
Radius: (3 ÷ 2) + (12² ÷ (8 × 3)) = 1.5 + 6 = 7.5 ft
Half angle: asin(6 ÷ 7.5) = asin(0.8) = 0.9273 rad
Central angle: 2 × 0.9273 = 1.8546 rad, or 106.26°
Arc length: 7.5 × 1.8546 = 13.91 ft
Face area: (7.5² ÷ 2) × (1.8546 − sin(1.8546)) = 28.125 × 0.8948 = 25.17 ft²
Volume per arch: 25.17 × 1.5 = 37.76 ft³
With 10% waste: 41.53 ft³, or 1.538 yd³
Cost at $150/yd³: 1.538 × 150 = $230.70
Just over a yard and a half of concrete, and the arc length comes in at nearly 14 ft. The contractor who cuts the formwork needs that arc length number more than the volume number. It tells them how long a piece of plywood to bend.
Five arch types are covered, so you’re not hunting for a separate formula for each curve.
Radius of curvature and central angle come out alongside volume, which is what a fabricator actually needs.
Arc length is reported separately, which matters for formwork and curved trim.
Metric and imperial units mix freely. A span in feet with a 3-inch rise computes correctly.
Cost accepts per cubic yard, per cubic foot, or per cubic meter, matching how suppliers quote.
Nothing runs on a server. Arch dimensions and prices stay on your device.
Sixteen currencies cover most regions without a manual conversion.
The breakdown tab exposes every intermediate value, which helps when reconciling against a hand calculation.
Multiply face area by depth, then by the number of arches. Face area depends on the arch type. A semicircle uses π × radius² ÷ 2. A segmental arch uses the circular segment formula. Add waste before ordering material.
Radius equals (rise ÷ 2) plus (span² ÷ (8 × rise)). A 10 ft span with a 2 ft rise gives (1) + (100 ÷ 16) = 7.25 ft. Shallow arches have large radii; flat arches have infinite radius.
Span is the total width of the opening. Rise is the height of the arch above its spring line, which is the level where the curve starts. Both are needed to define a segmental or elliptical arch.
Arc length equals radius times central angle in radians. A semicircle with a 5 ft radius has an arc length of about 15.71 ft. The calculator computes it for any arch type.
A gothic arch has two arcs meeting at a point. The equilateral version uses a radius equal to the span, with each side sweeping 60 degrees. It’s taller and more pointed than a semicircular arch.
Material cost depends on volume and price. A 10 ft semicircular arch with a 1 ft ring uses about 1.45 cubic yards of material. At $150 per cubic yard, that’s roughly $218 in material before waste and labor.
Ten percent is a safe default. Arches have more formwork cuts and more wasted material than straight walls. Curved brickwork with radial cuts pushes waste toward 15 percent.
Yes. Enter the dimensions, pick Brick Masonry as the material, and set the price per unit volume. The math is the same for any material; only the cost and construction method change.
No. Volume is purely geometric. A brick arch and a concrete arch of the same dimensions use the same cubic volume. The material type is for record-keeping and cost context.
An arch carries load by compression along the curve. A lintel is a straight beam that carries load in bending. Arches are more material-efficient for long spans; lintels are simpler to build.
ADVERTISEMENT
ADVERTISEMENT