Span
6 ft
Free arch geometry tool
Arch Calculator
Calculate span, rise, radius, arc length, central angle, and segment area for circular segment arches using practical layout inputs.
Arch geometry
Arch inputs
Span, rise, radius, and arc layout
Quick examples
Rise
1.5 ft
Radius
3.75 ft
Arc length
6.955 ft
Central angle
106.26 deg
Arch area
6.29 sq ft
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What is an arch calculator?
An arch calculator helps you work out the geometry of a curved opening or layout from common measurements like span, rise, and radius. It is useful when building or trimming arched doorways, window heads, masonry openings, curved forms, templates, and decorative finish work.
In practical jobs, people often know only part of the arch geometry. Sometimes you know the opening width and center rise. Other times you know the design radius from drawings. A strong arch tool should let you work from whichever values you already have.
This calculator focuses on circular segment arches, which are common in framing, finish carpentry, templates, and masonry layout.
Why span, rise, and radius all matter
The shape of an arch is controlled by its geometry. If the span changes, the curve changes. If the rise changes, the radius changes. If the radius changes, the arc length and total curve change too. These values are tied together, which is why an arch is easy to sketch but harder to lay out accurately by eye alone.
That matters when you need a physical result like a plywood template, bent trim, arched casing, curved fascia, concrete form, or masonry opening. Small geometry mistakes can show up quickly once you try to cut material or repeat the same arch across multiple openings.
A practical arch estimator is especially helpful when you need the arc lengthfor curved material or the segment area for finish, formwork, or infill planning.
Changing rise changes the curve
A flatter or taller arch changes the radius and the final arc length.
Useful for real layout work
The results help when making templates, cutting trim, or laying out a curved opening.
Arc length helps with materials
Curved trim, fascia, and finish pieces usually need the actual arc length, not only the span.
Guessing creates mismatch
A small error in radius or rise can leave repeated arches looking visibly different.
How the arch calculation works
For a circular segment arch, the key values are span, rise, and radius. Once any valid combination is known, the rest of the geometry can be solved. The calculator then derives the central angle, arc length, and segment area.
Step 1: Start with a known geometry set
Use span and rise, radius and span, or radius and rise depending on what you already know.
Step 2: Solve the missing dimensions
The calculator derives the missing radius, rise, or span from circular-segment geometry.
Step 3: Find the central angle
Once the circle geometry is known, the central angle defines the size of the curved section.
Step 4: Return practical outputs
You get arc length, area, and layout values that are easier to use in the field or shop.
Core idea
Arch geometry is solved from span, rise, and radius of a circular segment.
Once those relationships are known, the arc length and segment area follow from the central angle of the circle.
This is why the calculator can switch between multiple input modes. Different projects start with different known values, but the same underlying geometry ties them together.
Quick reference examples for arch layout
These examples show how different inputs affect the curve and layout result.
| Situation | Why it matters |
|---|---|
| Same span, higher rise | The arch becomes taller and tighter, which changes both the radius and the arc length. |
| Same radius, wider span | The opening takes up more of the circle and the central angle increases. |
| Flat segment arch | A shallow rise often produces a much larger radius than people expect. |
| Trim or fascia work | Arc length is more useful than span when estimating curved material. |
| Template making | Radius and springline-to-center help when laying out the curve with shop or field methods. |
How to use this arch calculator
- 1
Choose the input mode that matches your project
Use span and rise for measured openings, or switch to a radius-based mode when the design radius is already known.
- 2
Enter the dimensions in one unit system
Use one consistent unit so the geometry reflects the actual opening or template dimensions.
- 3
Review the solved arch values
Use the radius, central angle, and springline values for layout and checking.
- 4
Use arc length for curved materials
Arc length is often the most practical output for curved trim, fascia, or finish pieces.
- 5
Use area only when segment area matters
The arch area is useful for infill, formwork, finish coverage, or other area-based planning.
Real-world uses, edge cases, and limitations
Useful for templates and trim
Helpful when laying out curved trim, casing, fascia, plywood forms, or decorative arch details.
Good for repeated openings
Useful when several arched openings need to match and you want consistent geometry.
Useful for curved material planning
Arc length gives a better material reference than span alone for curved finish or bending work.
Only for circular segment arches
Elliptical, gothic, parabolic, and custom multi-center arches need different geometry.
This calculator is most reliable for circular segment arches. If your project uses a non-circular arch profile, the results here will not match that different curve family.
Another practical limitation is finish build-up. Trim thickness, jamb build-out, plaster, drywall returns, and edge profiles can all slightly change the final visible arch compared with the raw framing geometry.
For exact fabrication work, it is still smart to use these values as the geometry basis and then confirm the actual template or mockup before cutting expensive material.
Frequently asked questions
- What is the span of an arch?
- The span is the straight horizontal distance across the opening from one spring point to the other.
- What is the rise of an arch?
- The rise is the height from the springline up to the highest point of the arch.
- Why is arc length different from span?
- Span is a straight-line width, while arc length follows the curve itself, so arc length is longer.
- What does the arch area represent?
- It is the segment area between the curved arch and the straight chord across the opening.
- Can I use this for an elliptical arch?
- No. This tool is for circular segment arches, not elliptical or other specialty arch profiles.
Solve arch geometry before you cut templates, trim, or curved materials
Use this arch calculator to solve span, rise, radius, central angle, arc length, and segment area for circular segment arches. It is built for practical layout, repeatability, and cleaner field or shop work.