SAG Calculator
Find the sag (sagitta) of a circular arc — the depth the curve drops from a straight line between its endpoints. Enter any two of chord, sag, and radius, and get the third plus the arc length and central angle. Handy for arches, curved trim, arcs, and layout. Includes a unit table, a by-sag table, and a downloadable PDF.
| Measure | Inches | Decimal ft | Ft-in |
|---|---|---|---|
| Enter two values to convert | |||
| Sag | Radius | Arc length | Angle |
|---|---|---|---|
| Enter a chord to compare | |||
How to use the SAG (sagitta) calculator
Enter any two of chord, sag and radius, choose the pair you know, and get the third value plus the arc length, central angle, diameter and sag-to-chord ratio — with a unit-conversion table, a by-sag comparison and a downloadable PDF.
Pick what you know
Choose a known pair — chord + sag, chord + radius, or radius + sag. The calculator finds the third.
Enter your measurements
Type the two values and set each unit to inches or feet. Chord + sag is the usual pair for an arch.
Add material (optional)
Enter how many identical arcs and a price per foot to get total curved length and cost.
Read the result & save a PDF
See radius, arc length, angle and diameter, compare by sag, and download it for the shop.
How arc sag is calculated
A circular arc is fixed by just two measurements. The sagitta — the depth of the curve — ties the chord, the sag and the radius together, and from any two the rest follow by the geometry of the circle.
Worked example — 48″ chord, 9″ sag (chord + sag)
Radius: (48² ÷ 8 + 9² ÷ 2) ÷ 9 = (288 + 40.5) ÷ 9 = 36.5 in
Half-chord: 48 ÷ 2 = 24 in
Central angle: 2 × arccos(27.5 ÷ 36.5) = 82.2°
Arc length: 36.5 × (82.2° in radians) = 52.4 in
Sag ratio: 9 ÷ 48 = 18.8% of chord
A 48-inch opening with a 9-inch rise rides on a 36.5-inch radius — cut to that radius, and you’ll follow about 52.4 inches of curve from end to end.
Arc sag reference charts
The relationships that come up most — how the radius and arc react to sag, how the sag-to-chord ratio reads as a shape, and quick radius picks for common openings.
| Sag (rise) | Radius | Arc length | Central angle | Shape |
|---|---|---|---|---|
| 4.5 in | 66.3 in | 49.1 in | 42.5° | Shallow / segmental |
| 9 in | 36.5 in | 52.4 in | 82.2° | Gentle arch |
| 12 in | 30.0 in | 55.6 in | 106.3° | Medium arch |
| 18 in | 25.0 in | 64.4 in | 147.5° | Deep arch |
| 24 in | 24.0 in | 75.4 in | 180° | Semicircle |
For a fixed chord, more sag means a tighter radius and a longer arc. A semicircle is reached when the sag equals half the chord — here, 24 inches.
| Sag ratio (sag ÷ chord) | Feel | Where it fits |
|---|---|---|
| Under 10% | Flat, subtle curve | Segmental arches, wide openings, large radius |
| 10–20% | Gentle, classic arch | Most door and window arches |
| 20–35% | Pronounced curve | Feature arches, curved trim |
| 50% | Full semicircle | Round-top windows, Roman arches |
The sag ratio is the quick way to picture an arch without doing the math each time — a smaller ratio is flatter with a bigger radius, a larger ratio is a tighter, deeper curve.
| Chord × sag | Radius | Arc length | Central angle |
|---|---|---|---|
| 36 in × 6 in | 30.0 in | 38.6 in | 73.7° |
| 48 in × 9 in | 36.5 in | 52.4 in | 82.2° |
| 60 in × 12 in | 43.5 in | 66.3 in | 87.3° |
| 72 in × 12 in | 60.0 in | 77.2 in | 73.7° |
| 96 in × 16 in | 80.0 in | 103.0 in | 73.7° |
Values rounded. Arc length is the curved distance to cut or bend — always a little longer than the straight chord.
An arc is a balance of three measurements: the chord sets the width, the sag sets the depth of the curve, and together they fix the radius of the circle it rides on. The arc is the real curved length from end to end.
- Any two fix the third — give it chord and sag, chord and radius, or radius and sag, and the rest follow.
- More sag, tighter radius — deepen the curve on the same chord and the radius shrinks while the arc grows.
- Arc beats the chord — the curved length is always longer than the straight span, so cut to the arc.
Everything the calculator works out
One pair of measurements gives you the missing value, the arc length, the central angle, the diameter and the sag ratio — plus unit conversions, a by-sag comparison and a report for the shop.
The figures behind every arc
Built for anyone laying out a curve from flat measurements
An arched doorway, a run of curved trim, a radius wall, a template to cut — the chord-sag-radius relationship is the same, and knowing the radius and arc length before you cut keeps every piece true.
Framing an arched doorway or window and needing the radius to mark and cut the curve from the span and rise.
- Work from chord + sag
- Set a jig to the radius
- Cut to the arc length
Running curved molding or casing over an arch and ordering the right length of material for one or many pieces.
- Use the arc length to buy
- Multiply across identical arcs
- Add margin for joints
Laying out a brick, stone or steel arch where the radius and central angle set the whole form over an opening.
- Fix the radius first
- Check the central angle
- Confirm structural with an engineer
7 tips for getting arc sag right
The small measuring and layout habits that keep a curve true — and the checks that stop a piece coming out short.
Arc sag FAQ
The geometry and layout questions people ask most when working a curve from chord, sag and radius.
For a circular arc, radius = (chord² ÷ 8 + sag² ÷ 2) ÷ sag, with the chord and sag in the same units. The answer comes out in those units.
For a 48-inch chord with a 9-inch sag: (48² ÷ 8 + 9² ÷ 2) ÷ 9 = 36.5 in. Once you have the radius you can set a trammel or jig to mark the curve, and the calculator also returns the arc length and central angle from the same two inputs.
The sagitta, or sag, is the depth of a curve — the straight-line distance from the middle of the chord to the arc itself. The chord is the straight line joining the arc’s two endpoints, and the sag is how far the curve rises or drops from that line at its midpoint.
On an arch it’s simply the rise at the center. Sag, chord and radius are locked together by the circle’s geometry, so knowing any two of them fixes the third, which is what the calculator solves.
It’s an aesthetic choice, measured by the sag-to-chord ratio. Shallow segmental arches often sit below about 0.15, a gentle arch lands around 0.2, and a ratio of exactly 0.5 — where the sag equals half the chord — gives a full semicircle.
A flatter arch needs a much larger radius and looks subtle; a deeper one tightens the radius and reads as a strong curve. The calculator shows the sag ÷ chord ratio so you can compare options before you commit to a radius.
Arc length is the radius multiplied by the central angle in radians, where the central angle is 2 × arccos((radius − sag) ÷ radius). It’s the true curved distance from end to end — the amount of trim or material you’ll follow along the edge.
For a 48-inch chord with a 9-inch sag, the arc length is about 52.4 inches, compared with the 48-inch straight chord. The calculator computes this from whichever two values you enter, so you never have to work the trigonometry by hand.
They’re three views of the same circular arc. The chord is the width across, the sag is the depth of the curve from that chord, and the radius is the radius of the circle the arc belongs to. Fix any two and the third is determined.
That’s why the calculator offers three modes — chord plus sag, chord plus radius, or radius plus sag. Whichever pair you can measure on site, it finds the rest, along with the arc length, central angle and diameter.
Yes. Measure the chord (the span of the opening) and the sag (the rise), and the calculator gives you the radius to cut or bend to and the arc length to buy. The number-of-arcs field multiplies that length across several identical pieces, and a price per foot estimates cost.
Use the arc length as your minimum material and add extra for cuts, joints and fitting. For a run of matching arches, working from one chord and sag keeps every piece to the same radius.
Start from the arc length, which is the real curved distance along the piece — always longer than the straight chord. For several identical arcs, multiply the arc length by the quantity to get the total curved length.
Then add a margin for saw kerf, joints, fitting and trimming, since it’s easy to cut a curved piece shorter but impossible to add length back. The calculator’s total curved length and optional cost give you a buy figure to start from.
The geometry is exact for a circular arc — the radius, arc length and angle are correct for the chord and sag you enter. What it does not account for is structural load, span limits, material strength or connection details.
For a load-bearing arch — masonry, a timber or steel arch, anything carrying weight — use this to plan the shape, then confirm the design and materials with a qualified engineer and follow the relevant building code.
Estimate Only — Geometry, Not an Engineered Design: This calculator uses exact circular-arc geometry — sag = radius − √(radius² − (chord ÷ 2)²), and radius = (chord² ÷ 8 + sag² ÷ 2) ÷ sag — and assumes a true circular arc with the chord and sag measured accurately. It does not account for structural load, span limits, material strength, springback when bending, or connection and support details. Use the radius, arc length and central angle for layout, cutting and material estimates, and always add margin for kerf, joints and fitting. For masonry, timber or steel arches, or any load-bearing or life-safety span, follow the material and product specifications and consult a qualified engineer where required by code. For reference and estimating only; verify every structural arch against the manufacturer’s specs and an engineer.