Arched doorways, windows, bridges and garden arbours are usually circular segments or parabolas. To lay one out you need the radius (for a string or trammel) or the shape of the curve, and to order trim or flexible moulding you need the length along the arc.
How to use the Arch Calculator
- Arch type
- Span
- Rise
- Every result field is editable: type into Radius (circular) and Rise 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
Choose the type of arch and enter the span (width at the base) and the rise (height of the arch above the springing line). The calculator returns the radius and angle for a circular arch, the focus for a parabolic arch and, for both, the arc length and the area under the arch.
Tips for an accurate estimate
- Take measurements at floor level and at the ceiling – walls and rooms are often slightly out of square.
- Use the manufacturer’s coverage, spacing and installation data when it differs from the typical values here.
- Keep a little extra of every finish material for touch-ups and repairs later.
- Check local building codes and permits before structural, electrical, plumbing or gas work.
Common mistakes to avoid
- Sizing heating or cooling equipment by area alone when insulation, windows and climate make a big difference.
- Buying too little paint, stain or flooring and ending up with mismatched batches.
- Overlooking the extra material for steps, railings, gables and trim.
Inputs explained
| Field | What to enter |
|---|---|
| Arch type | Choose one: Circular (segmental), Parabolic |
| Span | Units: in, cm, mm, ft, m |
| Rise | Units: in, cm, mm, ft, m |
What each result means
- Radius (circular) = ([Span]^2÷4 + [Rise]^2)÷(2×[Rise])
- Central angle (circular) = deg(4×atan(2×[Rise]÷[Span]))
- Focus below apex (parabolic) = [Span]^2÷(16×[Rise])
Example values
How the main result changes when Rise changes and everything else stays at the example values above:
| Rise (in) | Radius (circular) (in) |
|---|---|
| 6 | 51 |
| 9 | 36.5 |
| 12 | 30 |
| 15 | 26.7 |
| 18 | 25 |
| 24 | 24 |
Formula
Circular: R = (s²/4 + h²) ÷ 2h; central angle θ = 4 × atan(2h ÷ s); arc length = R × θ; area = R²/2 × (θ − sin θ)Parabolic: area = ⅔ × s × h; focus f = s² ÷ 16h; arc length = a√(1 + (2h/a)²) + a² asinh(2h/a) ÷ 2h, a = s/2Worked example
Inputs — Arch type: Circular (segmental); Span: 48 in; Rise: 12 in.
Results — Radius (circular) = 30 in; Central angle (circular) = 106.26 °; Arc length = 55.638 in; Area under the arch = 402.57 in²; Focus below apex (parabolic) = 12 in.
Key terms
- Springing line
- The level where the arch curve begins.
Frequently asked questions
How do I find the radius of an arch?
R = (span²/4 + rise²) ÷ (2 × rise). A 48 in span with a 12 in rise has a 30 in radius.
What is a semicircular arch?
One whose rise is half the span; its radius equals the rise.
How do I draw a parabolic arch?
Mark the apex and the two springing points and plot y = h(1 − (2x/s)²), or use a string and nails at the focus.
What formula does the Arch Calculator use?
Circular: R = (s²/4 + h²) ÷ 2h; central angle θ = 4 × atan(2h ÷ s); arc length = R × θ; area = R²/2 × (θ − sin θ).