Lancet Arch Calculator

Name: Lancet Arch - Geometry Calculator
Author: Jürgen Kummer

Calculations at a lancet arch. This is a widthwise bisected pointed oval. The width must be smaller than twice the height. Lancet archs are an important element in Gothic architecture for windows and doors.
Enter two values at arch height, base width and radius of one of the original circles and choose the number of decimal places. Then click Calculate.

Formulas:

h = \sqrt{r \cdot b - \frac{b^{2}}{4}}

b = 2 \cdot (r - \sqrt{r^{2} - h^{2}})

r = \frac{4 h^{2} + b^{2}}{4 b}

p = 2 r \cdot \arccos [1 - \frac{b}{2 \cdot r}] + b

A = \frac{r \cdot (p - b)}{2} - h \cdot (r - \frac{b}{2})

Height, width, radius and perimeter have the same unit (e.g. meter), the area has this unit squared (e.g. square meter).

A lancet arch has three corners. Such shapes can therefore also be called triangles, although not with the usual mathematical meaning that a triangle consists of three points connected by straight lines. The pointed arch has this in common with the interarc triangle and the circular arc triangle, which also have three corners but not three straight lines.
The lancet arch has one axis of symmetry, passing through its apex and the center of the base. Otherwise, it has no symmetries.

Lancet arches are a variety of the more comprehensive shape pointed arches. The earliest known pointed arches appeared in a primitive form among the Sumerians around 5,000 years ago. They later became an element of Islamic architecture. They spread in Europe during the Gothic period. Pointed arches can have more complex shapes than the lancet form, but this bisected pointed oval is quite common. This lancet arch then often forms the upper part of a door or window and is then continued downwards by two parallel straight lines perpendicular to the floor.

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