Arch Calculator

Calculate arch radius, rise, and curve length

This arch calculator helps you plan the dimensions of a curved opening or arch. Enter the span, which is the clear width between the spring points, and choose a semicircular, segmental, or elliptical shape. For a segmental or elliptical arch, enter the rise from the spring line to the highest point. The calculator estimates radius, arch curve length, opening area, and approximate material volume from the arch thickness and depth.

These measurements are useful for early design, checking a drawing, estimating trim or masonry, and comparing arch proportions. Imperial and Metric units are supported. The diagram gives a simple view of the arch shape and highlights the span and rise used in the estimate.

An arch calculator does not design a load-bearing structure. Masonry arches, lintels, and structural openings must be designed for their loads, supports, materials, and local requirements. Use these results for geometry and planning only, and have structural designs reviewed by a qualified professional.

Arch terms: span, rise, radius, and spring line

Span is the horizontal distance across the opening from one spring point to the other. The spring line is the level where the curved portion begins at each side. Rise is the vertical distance from that line to the crown, the highest point of the arch. Radius describes the circle that forms a circular arch. These dimensions work together to define the shape.

An arch with a larger rise relative to its span looks taller and more curved. A shallow segmental arch has a small rise compared with its width. A semicircular arch has a rise equal to half its span. An elliptical arch is flatter near the sides and has a different curve from a circle, even when its span and rise match a circular segment.

How to use the arch calculator

Select feet or metres and choose an arch type. Enter the clear opening span. A semicircle calculates its rise as half of the span automatically. For a segmental arch, enter the measured rise; for an elliptical arch, enter the vertical rise that defines the ellipse with the span.

Enter arch thickness and wall depth to estimate the material band area and volume. Thickness is measured radially for circular arches and used as an approximate band thickness for elliptical arches. Wall depth is the distance through the wall or material. If you only need the curve dimensions, you can still use the geometric results without relying on the volume estimate.

Review the calculated radius, curve length, opening area, and outer curve length. The radius for an elliptical arch is shown as a crown curvature radius and a spring curvature radius because an ellipse does not have one constant radius. Use the appropriate drawing dimension for the type of arch being built.

Arch geometry formulas

For a circular segment, the radius can be found from the span and rise. The central angle then determines the arc length. A semicircular arch is a special case where the radius is half the span and the central angle is 180 degrees. Elliptical geometry uses horizontal and vertical semi-axes, and this calculator estimates the half-ellipse arc length with a standard approximation.

• Semicircle radius = span ÷ 2; rise = span ÷ 2
• Circular segment radius = (span² + 4 × rise²) ÷ (8 × rise)
• Circular segment angle = 4 × arctangent(2 × rise ÷ span)
• Circular arch arc length = radius × central angle in radians
• Circular segment opening area = radius² ÷ 2 × (angle − sine of angle)
• Elliptical semi-axes = span ÷ 2 and rise
• Approximate elliptical arch length = half the ellipse circumference

Worked example: a segmental arch

Imagine a segmental arch with an 8-foot span and a 2-foot rise. The radius is (8² + 4 × 2²) ÷ (8 × 2), which equals 3 feet. The central angle is about 166 degrees. Multiplying the radius by the angle in radians gives an inner curve length of roughly 8.7 feet.

The opening area below the curve is about 11.1 square feet. If the arch band is 8 inches thick and the wall is 8 inches deep, the calculator uses the ring geometry to estimate a material volume. Actual brick, mortar, stone, or formwork quantities can differ because of joint thickness, cuts, support details, and construction tolerances.

The same span with a 4-foot rise would make a semicircle with a 4-foot radius and a half-circle arc length of about 12.6 feet. A taller curve needs a longer arch ring. This is why span alone is not enough to estimate curve length or material quantity.

Semicircular arch dimensions

A semicircular arch is one half of a circle. Its radius is exactly half the span, and its rise equals the radius. For a 6-foot span, the radius and rise are both 3 feet. The curved length is half the circle circumference, or pi multiplied by the radius. This makes semicircle geometry quick to lay out with a center point and compass or trammel.

Check whether a drawing calls for the clear inside opening, the outside face, or the arch ring thickness. If the span refers to the outer edge rather than the clear opening, the radius you calculate will describe a different curve. Measure between the correct spring points and label inner and outer dimensions clearly.

Segmental arch radius and rise

A segmental arch is a circular arc that is less than a semicircle. It is usually wider and flatter than a half-circle, with a rise smaller than half the span. Given its span and rise, the calculator finds the circle radius and angle. A shallow arch has a larger radius because it is a small part of a much larger circle.

For layout, the radius can help locate the circle center below the spring line. The exact center position is important when transferring a curve to a template or form. Confirm the rise at the crown and the spring line on both sides; small errors can change the curve noticeably across a wide opening.

Elliptical arch radius and curve length

An elliptical arch uses a half ellipse instead of a circular arc. Its horizontal semi-axis is half the span, and its vertical semi-axis is the rise. The curve does not have a single radius. This calculator reports approximate radii of curvature at the crown and spring points to help describe how the ellipse changes through the curve.

The half-ellipse arc length is estimated with Ramanujan's circumference approximation. It is useful for material planning, but a plotted curve or CAD drawing may be preferable for a precise template. Elliptical arches can be useful when a flatter crown or a particular visual proportion is needed, but the shape should be clearly called out in the design.

Arch thickness, depth, and material volume

Arch thickness is the width of the ring following the curve, measured from the inner curve toward the outer curve. Wall depth is the thickness of the wall through the opening. The calculator estimates the ring area and multiplies it by depth to produce an approximate volume. For circular arches it uses an annular sector; for ellipses it approximates the curved band as arc length times thickness.

The volume is not an exact brick count or mortar takeoff. Masonry units have specific sizes and joints, while stone can have irregular shapes and waste. Concrete arches require formwork and reinforcement details. Use the volume to compare concepts, then prepare a material layout from actual units and construction details.

Opening area under an arch

Opening area is the area between the spring line and the inside curve. This calculator reports the circular segment area for semicircular and segmental shapes, or the half-ellipse area for an elliptical arch. The value can help estimate glazing, infill, or the open portion of a façade when the rest of the opening is rectangular below the spring line.

If your opening includes vertical sides below the arch, add that rectangle separately for a complete opening area. Do not confuse the arch segment area with the total area of a door or window opening. For an irregular opening, split it into simple geometric shapes and calculate each section.

How to measure an existing arch

Measure the clear span between the points where the curve begins. Establish the spring line with a level and measure vertically to the crown for rise. If the arch is covered by trim or a finish, determine whether you need the finished opening or the structural opening. Measure thickness at more than one point if the curve or material is irregular.

For an existing arch, do not assume its curve is a perfect circle. Compare several points along the curve with the calculated shape or make a full-size template. A shallow circular segment and an ellipse can look similar near the crown but differ toward the sides. Record reference points and dimensions before removing trim or making a replacement piece.

Common arch measuring and design mistakes

One common error is using the overall outside width as the clear span. Another is measuring rise from the floor or opening base instead of the spring line. Radius formulas require consistent span and rise measurements from the same arch face. Keep all dimensions in the selected unit and distinguish inside dimensions from outside dimensions.

A second mistake is treating an ellipse as a circle because both shapes have a curved top. An elliptical arch has changing curvature, so a circular radius cannot describe its full curve. Also, the calculated radius is not a structural capacity rating. It does not tell you whether the arch can support a wall, roof, or load above it.

Finally, do not use a volume estimate as a purchase list without accounting for joints, cuts, support conditions, and waste. Build a material layout from the actual brick, block, stone, concrete, or trim dimensions.

Using an arch template or full-size drawing

A full-size drawing can make arch construction easier to check. Mark the spring points, span, crown, rise, and inner curve on a stable surface or template. For a circular segment, locate the calculated center and strike the curve with a string or trammel of the correct radius. For a semicircle, the center lies on the spring line midway between the ends.

For an elliptical curve, use a plotted drawing, CAD layout, or a suitable ellipse construction method rather than a single-radius compass. Check that the curve passes through the intended spring points and crown. Templates should be reviewed before cutting valuable material or building permanent formwork.

Arch materials and construction notes

Different arch materials need different detailing. Brick and stone arches rely on their shape, support, mortar joints, and surrounding construction. Concrete arches need suitable forms and reinforcement as designed. Wood or metal trim may be non-structural, but it still needs correct attachment and finish details. This calculator only describes geometry and approximate quantities.

For load-bearing work, the arch, abutments, foundations, and material strengths must be designed together. A qualified engineer or architect can specify the profile and support conditions. Follow local building requirements and manufacturer instructions, especially where the arch carries roof, floor, or masonry loads.

Limits of this arch calculator

This tool calculates idealized semicircular, circular segment, and elliptical arch geometry from span and rise. It does not model irregular curves, settlement, joints, material strength, loads, abutments, foundations, reinforcement, construction tolerances, or code requirements. Elliptical arc length and band volume are approximate. The diagram is an illustration and is not a structural drawing.

Verify dimensions on the project drawing and at the site. For structural arches, obtain a professional design review. For fine trim work or CNC cutting, use a full-scale CAD profile or template and confirm the curve against the intended material and installation method.

Arch planning checklist

Before laying out an arch, confirm the shape, clear span, spring line, rise, inner face, outer thickness, and wall depth. Check whether the curve is circular or elliptical and whether dimensions refer to the finished opening or structural construction. Make sure the unit system matches every measurement.

Compare the curve length and volume with the actual material size, joint pattern, and cut plan. Prepare a drawing or template and verify it before fabrication. For structural work, have the load path and supports checked by a qualified professional. Clear dimensions and a reviewed profile make it easier to communicate the design to builders and suppliers.