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Arc and Chord Calculator

Radius and arc length for an arched opening.

Your measurementsEvery assumption editable01

The straight width across the opening.

How far the arc rises above the chord at its centre.

Arc geometryCalculated live02

Result

24.25in radius

24.25 in radius — 40.568 in of arc

Chord
36 in
Rise
8 in
Radiusswing the trammel from this
24.25 in
Included angle
95.85°
Arc lengthmaterial needed along the curve
40.5678 in
Centre below the chordhow far down to set the trammel point
16.25 in

Chord and rise are the two dimensions you can actually measure on site — the opening width and how high you want the curve. Radius and centre point follow from them, which is the reverse of how geometry textbooks pose it and the right way round for building.

The centre-below-chord figure is what you need in practice. It tells you where to put the trammel point or the nail, which is usually somewhere below the opening rather than anywhere convenient.

Arc length is the material the curve consumes, and it is always more than the chord. On a laminated or kerfed arch that difference decides how much stock to prepare.

Span and rise are the two things you can actually measure on site — how wide the opening is and how high you want the curve. Radius and centre point follow from them, which is the reverse of how textbooks pose it and the right way round for building.

Why use this tool?What it does differently03

Why use this tool?

From what you can measure

Span and rise, rather than radius and angle, which are what you are trying to find.

Trammel centre

How far below the chord to set the pivot — usually somewhere inconvenient, which is why knowing beats guessing.

Arc length

The material the curve consumes, always more than the span.

Included angle

For laying out segments or checking against a template.

How this worksThe method04

How this arc and chord calculator works

Radius comes from half the chord squared plus the rise squared, all divided by twice the rise. That single expression turns two site measurements into the radius you need to swing.

The centre sits radius-less-rise below the chord line. For a shallow arch that is a long way down — a 36 in opening with an 8 in rise has its centre 16.25 in below the springing line, which is often below the floor.

Arc length is twice the radius times the half-angle in radians. It is always longer than the chord, and on a laminated or kerfed arch that difference decides how much stock to prepare.

How to use itStep by step05

How to use it

  1. Step 1: Measure the span

    The straight width across the opening — the chord.

  2. Step 2: Decide the rise

    How high the curve stands above the chord at its centre.

  3. Step 3: Set the trammel

    At the calculated distance below the chord, and swing the radius.

  4. Step 4: Prepare for the arc length

    Not the span. The curve consumes more material than the opening is wide.

Example usageWorked figures06

Example usage

A segmental arch
A 36 in opening with an 8 in rise needs a 24.25 in radius, swung from a point 16.25 in below the chord. The arc itself is 40.568 in long.
A shallow curve
The same 36 in span with only a 3 in rise needs a 55.5 in radius — nearly five feet, from a centre well below the floor. Shallow curves need long trammels.
A semicircle
A rise equal to half the span gives a radius of exactly half the span and a centre right on the chord line. That is the maximum this geometry describes.
Frequently asked questionsCommon questions07

Frequently asked questions

How do I find the radius of an arch?

Half the span squared plus the rise squared, divided by twice the rise. For a 36 inch span with an 8 inch rise the radius is 24.25 inches.

Where do I put the trammel point?

Radius less rise below the chord line. On a shallow arch that can be several feet down, often below floor level, which is worth knowing before you start.

How much material does the curve need?

The arc length, which is always more than the span. A 36 inch opening with an 8 inch rise has an arc of 40.568 inches.

What is a chord?

The straight line across the ends of the arc — the span of the opening. The rise is measured from the middle of that line up to the curve.

Can the rise exceed half the span?

Beyond half the span the curve passes a semicircle and the geometry changes character. This handles up to a semicircle, which covers essentially all architectural arches.

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