Arch Calculator - Calculate Radius, Span & Rise

What Is an Arch Calculator?

An arch calculator finds the dimensions of a circular arch — its radius, span, and rise — from any two of those values. It also works out the arc length and the arch area. Architects, builders, and hobbyists use it to plan doorways, windows, bridges, and garden features shaped like a curve.

Key Arch Dimensions

A circular arch is a piece cut from the edge of a circle. Five measurements describe its shape:

• Radius: the distance from the center of that circle to any point on the arch's curve.
• Span: the straight horizontal distance between the two bottom ends of the arch, also called the width.
• Rise: the vertical distance from the springing line (the flat line joining the two ends) up to the highest point of the arch.
• Arc length: the distance along the curved line of the arch, from one end to the other.
• Arch area: the area of the curved region enclosed between the arch and the springing line.

Arch Dimension Formulas

These formulas apply to a circular arch, one built from a segment of a single circle. Most doorway, window, and bridge arches have a rise no greater than the radius. The calculator also handles taller horseshoe arches, where the rise is more than the radius but no more than twice the radius. In a horseshoe arch the curve is more than half a circle, so the ends turn back inward.

Rise formula (from radius and span)

$\text{Rise} = \text{Radius} - \sqrt{\text{Radius}^2 - \left(\frac{\text{Span}}{2}\right)^2}$

Valid when radius and span are positive and span ≤ 2 × radius. This formula gives the lower of the two arches that fit a given radius and span, so the rise it returns is never more than the radius. The calculator works it out in an equivalent rearranged form, (Span/2)² ÷ (Radius + √(Radius² − (Span/2)²)), which stays accurate for very shallow arches.

Radius formula (from span and rise)

$\text{Radius} = \frac{\text{Span}^2}{8 \times \text{Rise}} + \frac{\text{Rise}}{2}$

Valid when span and rise are positive.

Span formula (from radius and rise)

$\text{Span} = 2 \times \sqrt{2 \times \text{Radius} \times \text{Rise} - \text{Rise}^2}$

Valid when radius and rise are positive and rise ≤ 2 × radius. A rise above the radius gives a horseshoe arch. A rise of exactly twice the radius closes the circle, and the span becomes 0.

Arc length formula

$\text{Arc Length} = \text{Radius} \times \theta$

Here θ (theta) is the central angle, in radians, that the arch's curve traces out at the circle's center. The calculator finds it with the two-argument arctangent, atan2, which works for every arch shape, including horseshoe arches where θ is more than π:

$\theta = 2 \times \operatorname{atan2}\left(\frac{\text{Span}}{2},\ \text{Radius} - \text{Rise}\right)$

When the rise is no more than the radius, this gives the same angle as the simpler form:

$\theta = 2 \times \arcsin\left(\frac{\text{Span}}{2 \times \text{Radius}}\right)$

Arch area formula

$\text{Arch Area} = \frac{1}{2} \times \text{Radius}^2 \times (\theta - \sin\theta)$

This is the area of the pie-slice sector minus the triangle between its two radii. Written out, it is the same as:

$\text{Arch Area} = \frac{1}{2} \times \text{Radius}^2 \times \theta - \frac{1}{2} \times \text{Span} \times (\text{Radius} - \text{Rise})$

For a horseshoe arch, Radius − Rise is negative, so the triangle term is added instead of subtracted.

How to Use the Arch Calculator

The calculator offers three modes, one for each dimension you might be missing.

Calculate rise, when the radius and span are known: enter the radius and span. The tool returns the rise, arc length, and arch area.

Calculate radius, when the span and rise are known: enter the span and rise. The tool returns the radius, arc length, and arch area.

Calculate span, when the radius and rise are known: enter the radius and rise. The tool returns the span, arc length, and arch area.

The calculator rejects inputs that cannot form a real arch. The span cannot exceed twice the radius, the rise cannot exceed twice the radius, and every dimension must be a positive number. It shows an error message when a value breaks one of these rules.

Worked Examples

Example 1: Doorway Arch

A doorway arch has a span of 36 inches and a rise of 12 inches.

Radius = (36² ÷ (8 × 12)) + (12 ÷ 2) = (1296 ÷ 96) + 6 = 13.5 + 6 = 19.5 inches

θ = 2 × arcsin(18 ÷ 19.5) = 2 × arcsin(0.9231) = 2 × 1.176 = 2.352 radians

Arc Length = 19.5 × 2.352 = 45.86 inches

Arch Area = (0.5 × 19.5² × 2.352) − (0.5 × 36 × (19.5 − 12)) = 447.18 − 135 = 312.18 square inches

Example 2: Garden Arch

A garden arch has a radius of 4 feet and a span of 6 feet.

Rise = 4 − √(4² − (6 ÷ 2)²) = 4 − √(16 − 9) = 4 − 2.6458 = 1.35 feet

θ = 2 × arcsin(3 ÷ 4) = 2 × arcsin(0.75) = 2 × 0.848 = 1.696 radians

Arc Length = 4 × 1.696 = 6.78 feet

Arch Area = (0.5 × 4² × 1.696) − (0.5 × 6 × (4 − 1.3542)) = 13.57 − 7.94 = 5.63 square feet

Example 3: Bridge Arch

A bridge arch has a span of 50 feet and a rise of 15 feet.

Radius = (50² ÷ (8 × 15)) + (15 ÷ 2) = (2500 ÷ 120) + 7.5 = 20.83 + 7.5 = 28.33 feet

θ = 2 × arcsin(25 ÷ 28.33) = 2 × arcsin(0.8824) = 2 × 1.081 = 2.162 radians

Arc Length = 28.33 × 2.162 = 61.25 feet

Arch Area = (0.5 × 28.33² × 2.162) − (0.5 × 50 × (28.33 − 15)) = 867.67 − 333.33 = 534.34 square feet

Example 4: Horseshoe Arch

A decorative horseshoe arch has a radius of 10 inches and a rise of 15 inches. The rise is more than the radius, so the curve is more than half a circle.

Span = 2 × √(2 × 10 × 15 − 15²) = 2 × √(300 − 225) = 2 × √75 = 17.32 inches

θ = 2 × atan2(8.66, 10 − 15) = 2 × atan2(8.66, −5) = 4.189 radians (240°)

Arc Length = 10 × 4.189 = 41.89 inches

Arch Area = 0.5 × 10² × (4.189 − sin 4.189) = 50 × (4.1888 + 0.8660) = 50 × 5.0548 = 252.74 square inches

Common Uses for Arch Calculations

Builders use arch dimensions to cut formwork for concrete arches, lay out brick or stone templates for masonry arches, and plan the curvature of vaulted ceilings. Bridge designers use span, rise, and radius to balance load distribution against clearance. Restoration work relies on the same formulas to match the exact geometry of a historic arch when repairing or rebuilding it. Gardeners and DIY builders use them for arched trellises, gateways, and decorative openings.

This calculator covers circular arches only, ones formed from a single circle's arc. Elliptical, parabolic, Gothic, and catenary arches follow different curves and need different formulas.

A Short History of the Arch

Builders in ancient Mesopotamia used early arch-like forms as far back as 2500 BCE, though most were built by corbelling — stacking stones so each layer overhangs the one below — rather than a true curved arch. The Romans developed the semicircular arch into a standard building method, using it in aqueducts, bridges, and buildings such as the Colosseum. Medieval builders in Europe developed the pointed Gothic arch, which let them build taller, lighter structures. Later architects returned to semicircular forms based on precise proportions, and modern materials like steel and reinforced concrete now let arches span much greater distances.

Frequently Asked Questions

What is the difference between rise and height in an arch?

Rise is the vertical distance from the springing line to the top of the arch's curve. Height sometimes refers to the total height of an opening, including any straight wall below the springing line.

Does this calculator work for all arch types?

No. It works only for circular arches, ones shaped like a segment of a circle. Elliptical, parabolic, Gothic, and catenary arches follow different curves and need separate formulas.

What is the relationship between span and radius in a semicircular arch?

In a semicircular arch, the radius equals half the span, and the rise equals the radius. The arch forms exactly half of a circle.

Why can't the span be greater than twice the radius?

A span equal to twice the radius already produces a semicircle, the widest possible arch for that radius. A wider span would not fit on the circle at all.

Can the rise be greater than the radius?

Yes. A rise between the radius and twice the radius gives a horseshoe arch, which is more than half a circle. The rise cannot go past twice the radius, because that is the full height of the circle. In the calculate-rise mode the tool always returns the lower arch, so its rise is never more than the radius.

What units does the calculator use?

Any consistent unit works — inches, feet, meters, or centimeters. Enter every value in the same unit, and the results will be in that same unit, with area in that unit squared.