Arch Calculator

The Arch Calculator sizes a semi-elliptical arch from base length and rise. It returns focal point positions for the string method, approximate arc length, half-ellipse area, and eccentricity. Useful for doorway templates, trim work, and masonry layout.

Elliptical arch math

A semi-elliptical arch uses semi-major axis $a$ (half the base) and semi-minor axis $b$ (the rise). Focal distance from the center to each focus is:

$$f = \sqrt{a^2 - b^2}$$

To trace the arch with string and nails, set nails at both foci and at the top vertex. Tie a string of length equal to the full base ($2a$). Removing the top nail, draw the curve while keeping the string taut.

Drawing an elliptical arch

Draw the base line and mark its center.
Mark rise height perpendicular at the center.
Place focus nails at distance $f$ left and right of center.
Use a string equal to base length to trace the upper half of the ellipse.

Frequently Asked Questions

What is the difference between a Roman arch and an elliptical arch?

A Roman arch is a semicircle (half of a circle). An elliptical arch is half of an ellipse, allowing a lower rise over a wide span without excessive height.

Why use an elliptical arch for doorways?

Elliptical arches give a wide opening with a modest rise, fitting under standard ceiling heights better than tall semicircular arches.

How long should the tracing string be?

The string length equals the full base width of the arch ($2a$). That matches the constant sum of focal radii property of an ellipse.

What if rise is larger than half the base?

Then $b > a$ and the shape is not a valid horizontal semi-ellipse with the base as major axis. Reduce rise or widen the base.

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Arch Calculator

Arch Calculator

Elliptical arch math

Drawing an elliptical arch

Frequently Asked Questions

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