Arch Calculation Formula: Rise, Span, and Curve

An arch calculation formula relates an arch’s span, rise, and curve to describe its shape; for a simple circular arch, the radius can be calculated from the span and rise. Enter the span and rise in the same units to calculate the radius, which helps define the geometry but does not by itself determine whether an arch is structurally safe.

Formula

For a circular arc that meets a horizontal baseline at its two ends, the radius is r = (s² + 4h²) / (8h). Here, r is the circle’s radius, s is the horizontal span between the endpoints, and h is the vertical rise from the baseline to the arch’s crown. All three lengths must use the same unit. The calculator also reports the dimensionless ratio r/s; it is a shape comparison, not another length.

This formula assumes the arch is a circular segment: every point on the curve lies on the same circle. It does not apply automatically to pointed, parabolic, elliptical, or irregular arches. For a semicircle, the rise is half the span, so the radius is also half the span.

Worked Example

Using the calculator’s defaults, the span is 6 m and the rise is 1.5 m. Substitute these values:

1. Square the span: 6² = 36 m².
2. Square the rise and multiply by four: 4 × 1.5² = 9 m².
3. Add the results: 36 + 9 = 45 m².
4. Divide by eight times the rise: 45 ÷ (8 × 1.5) = 45 ÷ 12 = 3.75 m.

The circular-arch radius is 3.75 m. The radius-to-span ratio is 3.75 ÷ 6 = 0.625. Both results match the calculator’s displayed precision.

What the Measurements Mean

Span and rise describe the arch in a front view. The span is the straight, horizontal distance between the arch’s spring points—the places where the curved portion begins. It is not the length measured along the curve. The rise is the vertical distance from the straight line joining those spring points to the highest point of the arch.

• Span: the clear horizontal width between endpoints.
• Rise: the vertical height above the endpoint baseline.
• Radius: the distance from the circle’s center to the curve.
• Spring points: the two ends of the curved portion, not necessarily the outer edges of surrounding walls or supports.

A drawing should make clear which surfaces define the span. Measuring between the wrong faces can change the calculated geometry, even if the arithmetic is correct.

How to Use the Calculation

Measure the span and rise in the same units, then enter them as nonnegative lengths. The radius is calculated from those two dimensions. A larger rise relative to the span describes a deeper, more upright circular segment; a smaller rise describes a flatter one. The radius-to-span ratio offers a quick way to compare arches of different sizes, but it does not describe the full curve by itself.

For metric work, use metres or millimetres consistently. For US customary measurements, use feet or inches consistently; the formula works identically because the units cancel correctly to leave a length. For example, a span in feet and a rise in inches must not be entered together without conversion. Convert one value first, such as 6 ft = 72 in, then use inches for both.

The formula can help with preliminary geometry, drafting, or checking dimensions on a drawing. If you need the curved length of the arch rather than its radius, the radius alone is not the requested result. For a circular segment, the arc length also depends on the angle subtended by the curve at the circle’s center.

Choosing the Right Arch Model

First establish whether the curve is intended to be circular. A circular arch has a constant radius; its center may sit above or below the spring-point baseline depending on the shape. A three-centered arch uses multiple circular arcs, while a pointed arch consists of intersecting curves. A parabolic arch has a different mathematical profile. Applying the single-radius formula to those shapes gives, at best, an approximation for one part of the curve.

For a semicircle, the geometry is especially simple: rise = span ÷ 2 and radius = span ÷ 2. A shallower circular arch has a radius greater than half its span. The default example has a rise of one quarter of its span and gives a radius of 3.75 m, which is greater than the 3 m radius of a semicircle with the same 6 m span.

Common Errors and Limits

• Mixing units: Convert all dimensions to one unit before substituting. The formula cannot correct a feet-and-inches mismatch.
• Using the curve length as the span: Span is the straight horizontal chord between endpoints, not the distance along the arch.
• Measuring rise from the wrong baseline: Rise is measured from the line connecting the arch’s spring points.
• Using a zero rise: The formula divides by 8h, so h must be greater than zero. A flat line is not a circular arch segment with a finite radius under this calculation.
• Assuming geometry proves strength: The result says nothing about material, loads, foundations, connections, or construction quality.

Measurements and rounding affect the answer. If span or rise is approximate, the calculated radius is approximate too. Keep enough precision during calculation and round the final result to a sensible level for the measurement method; reporting extra decimal places does not make uncertain measurements more exact.

Frequently Asked Questions

What is the formula for an arch radius?
For a circular arch with span s and rise h, use r = (s² + 4h²) / (8h). Use the same units for span and rise.

How do I calculate an arch’s rise?
If you know the circular radius and span, rearrange the circle geometry: h = r − √(r² − (s/2)²) for the shallower circular segment. The radius must be at least half the span for this segment to exist.

Is an arch’s span the same as its width?
It often means the clear horizontal width between the arch’s spring points, but drawings and projects can define width differently. Check which endpoints the measurement is meant to cover.

Does this formula work for a pointed arch?
Not as a single-radius description. A pointed arch generally combines separate curves, so calculate each curve from its own geometry or use a method designed for that arch profile.

Can this formula tell me whether an arch is safe?
No. It calculates a geometric radius only. Structural safety requires information about loads, materials, supports, and construction, among other factors; geometry alone cannot establish it.

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in s | Span | m | 6 | 0
in h | Rise | m | 1.5 | 0
out r | Circular-arch radius | m | (s^2 + 4*h^2) / (8*h) | 2
out c | Radius-to-span ratio | - | r / s | 3