Gothic Arch Calculator: Build a Perfect Arch

A gothic arch calculator works out the height, width, and arc geometry of a pointed arch from its span and the radius of the two intersecting circles. You enter the span (the distance between the two base points) and the radius of each circular arc, and the calculator returns the arch height, the rise, the segment angles, and the total arc length. This is the same math used for cathedral windows, doorways, and garden trellises.

Formula

A true gothic arch (also called an equilateral or pointed arch) is formed by two circular arcs that meet at a point. Each arc is part of a circle with radius r, and the centers of those circles are located at the two base points of the arch. The span s is the distance between those centers.

The height of the arch from the base line to the point is:

h = r + √(r² – (s/2)²)

Where:

• s = span (horizontal distance between the two base points, in feet or meters)
• r = radius of each circular arc (same unit as span)
• h = total arch height from the base line to the tip (same unit)

The rise (the vertical distance from the base line to the point where the two arcs intersect) is simply the square root term: rise = √(r² – (s/2)²).

The angle of each arc segment (in degrees) is:

θ = 2 × arccos(s / (2r))

And the total arc length (the length of the curved line of the arch) is:

L = 2 × r × (θ × π / 180)

This assumes the two arcs are identical and symmetric. If you want a non-symmetric arch, you would need two different radii, but most gothic arches are symmetric.

Worked Example

Let’s use the default values from the calculator: span = 6 ft, radius = 4 ft.

1. Compute half the span: s/2 = 3 ft.
2. Square that: (s/2)² = 9 ft².
3. Square the radius: r² = 16 ft².
4. Subtract: 16 – 9 = 7 ft².
5. Take the square root: √7 ≈ 2.6458 ft (this is the rise).
6. Add the radius: h = 4 + 2.6458 = 6.6458 ft, which rounds to 6.65 ft.
7. For the angle: θ = 2 × arccos(3/4) = 2 × 41.4096° = 82.8192°, which rounds to 82.8°.
8. For the arc length: L = 2 × 4 × (82.8192 × π / 180) = 8 × 1.4455 = 11.564 ft, which rounds to 11.56 ft.

The calculator gives the same results: arch height 6.65 ft, rise 2.65 ft, angle 82.8°, arc length 11.56 ft.

When to Use a Gothic Arch Calculator

You would use this calculator whenever you need to lay out a pointed arch for construction, craft, or design. Common situations include:

• Building a gothic window frame or door header.
• Cutting plywood or drywall for an arched opening.
• Designing a garden trellis or arbor with a pointed top.
• Making a decorative arch for a stage set or craft project.
• Estimating the length of trim or molding needed to frame the arch.

The calculator gives you the exact dimensions you need to mark the curve on your material. Without it, you would have to guess or draw the arch freehand, which often leads to a lopsided result.

How to Choose the Radius

The radius you choose determines the shape of the arch. For a classic equilateral gothic arch, the radius equals the span. That gives a 60° angle at the base and a rise that is about 0.866 times the span. For a wider, flatter arch, use a radius larger than the span. For a taller, more pointed arch, use a radius smaller than the span—but be careful: the radius must be at least half the span, or the two circles won’t intersect above the base line.

If you want a specific rise, you can solve for the radius: r = ( (s/2)² + rise² ) / (2 × rise). But it’s often easier to pick a radius and see what height you get, then adjust.

Common Mistakes and Misconceptions

One common mistake is mixing up the rise and the total height. The rise is measured from the base line to the point where the arcs meet, while the total height includes the radius below the base line if the circle centers are below the base. In our calculator, the total height is rise + radius, because the center of each circle is at the base line. If you set the centers below the base line, the total height would be different.

Another mistake is using the span as the diameter of one circle. In a gothic arch, the span is the distance between the two circle centers, not the diameter. The radius is the distance from each center to the point where the arc meets the base, and that is the same as the radius of the circle.

A third misconception is that a gothic arch is a single curve. It is actually two separate arcs that meet at a point. The angle at the point is not 90°; it depends on the radius and span. For an equilateral arch (r = s), the point angle is 60°.

Limits and Real-World Considerations

The formula assumes perfect geometry and zero material thickness. In real construction, you will need to account for the thickness of the material you are cutting. For example, if you are cutting an arch out of plywood, the curve you mark is the centerline of the arch. The actual cut line will be offset by half the material thickness on each side.

Also, the calculator gives the arc length along the centerline. If you are bending a piece of trim or molding around the arch, you need the length along the outer edge, which will be longer. For a material of thickness t, the outer arc length is approximately L_outer = L + t × θ (with θ in radians).

For large arches, you may want to use a string and pencil to draw the curve directly. The calculator gives you the radius and center points, so you can set two nails at the base points and use a string of length r to draw each arc.

Alternative Arch Styles

If a gothic arch isn’t what you need, there are several other arch forms:

• Roman (semicircular) arch: a single half-circle; height equals radius, span equals diameter.
• Segmental arch: a shallow arc that is less than a semicircle; height is less than half the span.
• Elliptical arch: a half-ellipse; you need both a major and minor axis.
• Tudor arch: a four-centered arch, more complex to compute.

Each has its own formula, but the gothic arch is unique in that it is made of two intersecting circles, which gives it that distinctive pointed top.

Frequently Asked Questions

What is the difference between a gothic arch and a pointed arch? They are the same thing. “Gothic arch” is the common name, and “pointed arch” is the descriptive term. Both refer to an arch made of two circular arcs that meet at a point at the top.

Can I use this calculator for a door frame? Yes, but remember that the span is the width of the opening, and the height you get is the height from the base line to the point. You will need to add the height of the side walls if the arch starts above the floor.

What is the minimum radius for a given span? The radius must be at least half the span. If the radius equals half the span, the two circles just touch at the top, and the arch becomes a semicircle (a Roman arch). If the radius is less than half the span, the circles don’t intersect, so no arch is formed.

How do I draw a gothic arch on wood? Mark the two base points on your material. Set a nail at each point. Tie a string of length r to one nail, and draw an arc from the opposite base point upward. Repeat from the other nail. The two arcs will cross at the top. That crossing point is the tip of the arch.

Does the calculator account for material thickness? No. The calculator gives the centerline dimensions. For a real cut, you need to offset the curve by half the material thickness. For bending trim, use the outer arc length formula given above.

What if my arch is not symmetric? If the two radii are different, the formula changes. You would need to solve two circle equations simultaneously. This calculator assumes a symmetric arch, which is the standard for gothic arches.

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in span | Span (width) | ft | 6 | 0.1 | 100
in r | Radius of each arc | ft | 4 | 0.1 | 50
out h | Arch height | ft | r + sqrt(r^2 - (span/2)^2) | 2
out rise | Rise (above spring line) | ft | sqrt(r^2 - (span/2)^2) | 2
out ang | Segment angle (each side) | deg | 2 * acosd(span/(2*r)) | 1
out len | Total arc length | ft | 2 * r * (ang * pi / 180) | 2