# SAG Calculator (Sagitta)

URL: https://speedycalculator.com/calculators/construction/sag/
Last updated: 2026-10-25

> The SAG calculator finds the sagitta – the height of a circular arc above its chord – from the radius and chord length, plus the arc length and central angle. Edit the sag to find the radius.

## Inputs

- Radius (ft, m, in, yd, cm)
- Chord length (ft, m, in, yd, cm)

## Results

- Sagitta (sag)
- Arc length
- Central angle

## Formula

```
s = R − √(R² − (c/2)²)
Arc length = 2R × asin(c ÷ 2R)
Central angle = 2 × asin(c ÷ 2R)
```

## Explanation

The sagitta (from the Latin for arrow) is the depth of an arc measured at the middle of its chord. It is used to lay out curved walls, arches and bent trim, to check lens and mirror curvature and to find the radius of a curve you can only measure across. Carpenters often know the chord and the rise and need the radius to swing an arc.

### How to use the SAG Calculator

- **Radius**
- **Chord length**
- Every result field is editable: type into **Sagitta (sag)** and **Radius** plus every other field update to match.

Results update as you type, and every field accepts the units shown in its menu.

### How it works

Enter the radius and the chord length. The calculator returns the sag, the length along the arc and the angle the arc subtends. Type a sag value to work backwards to the radius.

### Tips for an accurate estimate

- Sketch the job and split irregular areas into rectangles, circles and triangles before you measure.
- Add a waste allowance – typically 5–10% for simple layouts and 15–20% for diagonal, curved or heavily cut work.
- Check the coverage or yield printed on the product, because sizes and densities vary between manufacturers.
- Buy all units of a finished material (tile, brick, siding) from the same batch so colours match, and keep a few spares for repairs.
- Confirm local prices – quarry, lumber and steel prices vary widely by region and season.

### Common mistakes to avoid

- Ordering exactly the calculated amount with no allowance for breakage, offcuts and settlement.
- Forgetting to subtract doors, windows and other openings – or subtracting small ones that still need cutting around.
- Confusing US short tons (2,000 lb) with metric tonnes (1,000 kg) when comparing quotes.

### Inputs explained

FieldWhat to enterRadiusUnits: ft, m, in, yd, cmChord lengthUnits: ft, m, in, yd, cm

### What each result means

- **Sagitta (sag)** = [Radius] - sqrt([Radius]^2 - ([Chord length]÷2)^2)
- **Arc length** = 2×[Radius]×asin([Chord length]÷(2×[Radius]))
- **Central angle** = deg(2×asin([Chord length]÷(2×[Radius])))

### Example values

How the main result changes when **Radius** changes and everything else stays at the example values above:

Radius (ft)Sagitta (sag) (in)5247.513.8691010.01812.57.887156.518204.849

## Worked example

Inputs — Radius: 10 ft; Chord length: 8 ft.

Results — **Sagitta (sag) = 10.018 in**; Arc length = 8.23 ft; Central angle = 47.16 °.

## Key terms

- **Chord**: A straight line joining two points on a circle.

## FAQ

**How do I find the radius from the sag?**
R = s/2 + c²/(8s), with s the sag and c the chord.

**What happens if the chord is longer than the diameter?**
It is impossible – a chord can be at most 2R, so the result is undefined.

**Where is the sagitta used?**
Arches, curved walls, lens design (sag of a surface), road and rail curves and woodworking templates.

**What formula does the SAG Calculator use?**
s = R − √(R² − (c/2)²).