# How to Calculate Geodesic Domes

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#### Things You'll Need

• Calculator

• Pencil

• Paper

• Balsa or basswood sticks

• Straight pins

Image Credit: NA/AbleStock.com/Getty Images

Geodesic domes were popularized by Buckminster Fuller in the 1950s. Since their introduction, geodesic domes have been constructed for many uses, including homes, containers, and structures for outer space. The name of the dome is from the chords of the structure that create great arcs, also known as geodesics. The dome's form is useful because it is approximately spherical and has a large volume relative to its surface area. Furthermore, the chords of the structure distribute loads around the interior volume, like a shell. There are many types of geodesic spheres, and each has unique geometric properties. The formulas for calculating most of the spheres are too entailed to include here, so use the references and resources provided to determine the construction specifications. Nevertheless, two very popular geodesic dome types are given below.

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## Step 1

Determine the purpose for the geodesic dome and what size the dome should be. Because the dome is spherical, a diameter or radius is an appropriate manner to describe the size.

After the size has been determined, find the desired type of geodesic dome from the references and resources. For simplicity, two types of dome are described here--icosahedral and truncated icosahedral. Both types are composed of regular polygons.

## Step 2

An icosahedron has 20 faces and is composed of equilateral triangles. Though it loosely approximates a sphere, the icosahedron is easy to construct and can incorporate many variations. An icosahedral geodesic dome omits 1, 5, or 15 faces from an icosahedron, depending on the desired form.

To calculate the chord length, determine the maximum exterior radius or the minimum interior radius of the polyhedron. The maximum exterior radius will give the size of the structure's footprint, and the minimum interior radius denotes the dome's usable volume.

Chord Length = Maximum Exterior Radius / 0.95106

Chord Length = Minimum Interior Radius / 0.75576

There is only one chord length for an icosahedral geodesic dome, so the calculations are complete.

A complete icosahedron has 20 faces, 30 chords, and 12 vertices or nodes.

## Step 3

A very popular form of geodesic dome is the truncated icosahedral geodesic dome. Apparent from its name, this geodesic dome type is created from a modified icosahedron. A truncated icosahedron has 32 faces, 90 chords, and 60 vertices or nodes. Unlike the icosahedron, the truncated icosahedron is made up of two shapes--regular hexagons and regular pentagons.

As with the icosahedral geodesic dome, the truncated icosahedral geodesic dome's chord length can be found relative to the radius.

Chord Length = Maximum Exterior Radius / 2.47801

Chord Length = Minimum Interior Radius / 2.42707

Though there is only one chord length for a truncated icosahedron, it is suggested the regular hexagons and pentagons are triangulated. The easiest way to do this is to construct the hexagons and pentagons with equilateral triangles. The hexagon will not be affected by the introduction of equilateral triangles, however the pentagons constructed with equilateral triangles will expand three-dimensionally, breaking the plane of the circumferential sphere. If this is not desired, a second chord length can be introduced to triangulate the pentagon with isosceles triangles. Triangles that will not break the plane of the pentagon will have the chord length:

Interior Pentagon Chord = Exterior Pentagon Chord / 1.17557

Otherwise, the chord lengths can approximate the shape of the sphere. The chord lengths within the hexagons and pentagons would be:

Interior Chord Length = Exterior Radius x [2 x sin ( Arc Angle / 2 )]

This formula will work for the chords with any geodesic form approximating a sphere.

## Step 4

After calculating the chords, test the calculations by making a balsa or basswood scale model of the geodesic dome. Use straight pins for the vertices or chord intersections. Remember the chords have been calculated as lines without dimensions. Find the depth of the connections, from the vertex, and multiply this dimension times 2. Subtract this from the calculated chord length, and this is the scaled length to be cut for the model.