Arch Calculator

The focal points (foci) of an ellipse are two fixed points inside the ellipse that define its shape. For an elliptical arch, these points are used with a string method to trace the perfect curve. The distance from the center to each focus is calculated as c = √(a² - b²), where a is the semi-major axis and b is the semi-minor axis. The closer the foci are to the center, the more circular the arch; the farther apart, the more elongated.

To draw an elliptical arch: First place two nails/pins at the focal points calculated by this tool. Tie a string loop that measures 2a (twice the semi-major axis) in total length. Loop the string around both pins and use a pencil inside the loop to trace the ellipse. The string method works because the sum of distances from any point on the ellipse to both foci equals 2a, creating the perfect elliptical curve every time.

Elliptical arches are commonly used in: Bridge design (stone arch bridges), windows and doorways (especially in classical architecture), garden trellises and arbors, fireplace openings, masonry arches in historic buildings, and modern architectural features. The elliptical arch distributes load efficiently while providing an elegant visual appearance that is less rounded than a semi-circular arch.

The semi-major axis (a) is half the total width of the arch, and the semi-minor axis (b) is half the total height. When a = b, the ellipse becomes a perfect circle. When a > b (wider than tall), the ellipse is horizontally elongated - this is the typical arch shape. The ratio a:b determines the eccentricity e = √(1 - b²/a²). An eccentricity near 0 means nearly circular; near 1 means very flat and elongated.