A Practical Guide to Determining Chord Lengths for Geodesic Domes
Geodesic domes, which Buckminster Fuller brought to prominence in the 1950s, take their name from the geodesic arcs formed by their structural chords. They have been built for purposes ranging from residences and storage units to space-based installations. Because the dome is nearly spherical and encloses a large volume relative to its surface, its chords spread loads throughout the interior much like a shell. Many geodesic sphere varieties exist, each with distinct geometric characteristics. Most require complex formulas beyond the scope of this guide, so references and resources should be consulted for construction specifications. Two widely used types are detailed here.
By Toby Young4 steps
Buckminster Fuller popularized geodesic domes during the 1950s. In the years since, people have built them as homes, containers, and even structures intended for outer space. The term "geodesic" comes from the structural chords that form large arcs, known as geodesics. A dome of this kind is useful because it is nearly spherical and has a large volume relative to its surface area, and its chords distribute loads around the interior volume in the manner of a shell. Geodesic spheres come in many varieties, each with its own geometric properties. The formulas for most of these spheres are too complex to include here, so consult the references and resources provided to establish construction specifications. Two particularly common types are covered below.
Step 1
First, decide what the geodesic dome will be used for and what size it needs to be. Since the dome is spherical, its size is appropriately expressed as a diameter or a radius.
Once the size is settled, select the specific type of geodesic dome from the references and resources. For clarity, two types are described here: icosahedral and truncated icosahedral. Regular polygons make up both types.
Step 2
An icosahedron consists of 20 faces, all equilateral triangles. It only roughly approximates a sphere, but it is simple to build and allows for many variations. An icosahedral geodesic dome leaves out 1, 5, or 15 faces from an icosahedron, depending on the form you want.
To find the chord length, first determine either the maximum exterior radius or the minimum interior radius of the polyhedron. The maximum exterior radius indicates the size of the structure's footprint, while the minimum interior radius indicates the dome's usable volume.
For the maximum exterior radius:
Chord Length = Maximum Exterior Radius / 0.95106
For the minimum interior radius:
Chord Length = Minimum Interior Radius / 0.75576
An icosahedral geodesic dome has only one chord length, so no further calculations are needed.
A complete icosahedron contains 20 faces, 30 chords, and 12 vertices or nodes.
Step 3
The truncated icosahedral geodesic dome is a highly popular form. As its name suggests, this type is based on a modified icosahedron. A truncated icosahedron has 32 faces, 90 chords, and 60 vertices or nodes. Unlike the icosahedron, it is built from two shapes: regular hexagons and regular pentagons.
Just as with the icosahedral geodesic dome, you can find the truncated icosahedral dome's chord length from the radius.
Chord Length = Maximum Exterior Radius / 2.47801
For the minimum interior radius:
Chord Length = Minimum Interior Radius / 2.42707
Even though a truncated icosahedron has only one chord length, triangulating the regular hexagons and pentagons is recommended. The simplest approach is to construct the hexagons and pentagons from equilateral triangles. Adding equilateral triangles does not affect the hexagon, but doing so to the pentagons will expand them three-dimensionally, breaking the plane of the circumferential sphere. If that is not wanted, a second chord length can be introduced to triangulate the pentagon using isosceles triangles. Triangles that do not break the plane of the pentagon will have this chord length:
Interior Pentagon Chord = Exterior Pentagon Chord / 1.17557
Alternatively, the chord lengths can approximate the shape of the sphere. The chord lengths within the hexagons and pentagons would then be:
Interior Chord Length = Exterior Radius x [2 x sin ( Arc Angle / 2 )]
This formula applies to chords in any geodesic form that approximates a sphere.
Step 4
After calculating the chords, verify the results by building a scale model of the geodesic dome from balsa or basswood. Use straight pins at the vertices or chord intersections. Keep in mind that the chords were calculated as lines without dimensions. Determine the depth of the connections from the vertex, multiply that dimension by 2, and subtract the result from the calculated chord length. The remaining value is the scaled length to cut for the model.
