Arch Calculator

How to use the elliptical arch calculator

What this calculator does

This calculator converts the two defining dimensions of a horizontal half-ellipse – the full base and the rise – into practical layout measurements. It locates the two foci, gives the string length for the traditional pins-and-string method, estimates the curved edge length, and produces coordinates for a template. It is a geometric layout aid, not a structural design check: it does not size masonry, framing, reinforcement, lintels, or foundations.

When to use it

Use it when laying out an elliptical doorway or window head, making a plywood or drywall template, checking whether a low-rise arch will fit below a ceiling, or placing focus nails for a full-scale string layout. The underlying focus relation is the standard ellipse identity described in the OpenStax explanation of ellipse equations and foci.

How to calculate

1. The calculator opens with a complete demonstration: an 88 in base and a 30 in rise. Its results and verified XLSX workbook are immediately available.
2. Replace Length of the arch (base) with the full opening width. Choose its unit beside the field.
3. Replace Height of the arch (rise) with the vertical distance from the spring line to the crown. Its unit can differ; the calculator converts it automatically.
4. Read the focus distance and focus-to-focus spacing, then use the profile and coordinate table to transfer the curve.
5. Select Download Excel to export the current typed inputs and results. Reset clears the demonstration data and disables export until both valid dimensions are entered again.

Input guide

Length of the arch (base) is required and must be a positive decimal number in inches, feet, millimeters, centimeters, or meters. A value such as 88 in means the entire horizontal opening, not the half-width. A larger base increases the semi-major axis, the string length, and usually the focal spacing. Do not enter a radius or mix a unit symbol into the text field; choose the unit from the adjacent list.

Height of the arch (rise) is required and uses the same available length units. A value such as 30 in is measured from the spring line to the highest point. For the horizontal ellipse modeled here, the rise must not exceed half the base after unit conversion. Increasing the rise makes the arch rounder and moves each focus toward the center. A rise exactly equal to half the base produces a semicircle with coincident foci at the center.

Output guide

Focus distance from center is the horizontal distance from the midpoint to either focus. Mark one focus this distance left and the other the same distance right. Focus-to-focus spacing is twice that value. String length equals the full major axis, so it matches the base for the ideal taut-string construction. Semi-major axis (a) is half the base; Semi-minor axis (b) equals the rise. Eccentricity is dimensionless: zero is a semicircle, while a value approaching one describes a progressively flatter ellipse. Approx. curved length estimates only the upper elliptical edge using a high-accuracy perimeter approximation; it is useful for trim planning but should allow for cuts and waste.

The Layout coordinate table lists each station from the left jamb, its signed offset from center, the calculated height above the base, and its percentage of the span. The profile uses exactly the same model values. It is a scaled analytical view, so use the table values for fabrication rather than measuring pixels on the screen.

Worked example

For the startup dimensions, the base is 88 in, so a = 44 in; the rise gives b = 30 in. The focal distance is c = √(a² – b²) = √(44² – 30²) = √1036 = 32.2490 in, displayed as 32.25 in. The two foci are therefore 64.50 in apart, and the string length is 88.00 in. At the center station, x = 0, so the ellipse equation gives y = b = 30.00 in, matching the crown height in the table and drawing.