List of numbers related to geometry

Introduction

If you come across a number and think it might be related to geometry, you may find the corresponding formula here. The approximate values are given to nine significant figures.

For polygons, these values are given:

Polygons on the list: regular polygons with sides of length one and 3–10, 12, 15, 16, 20, 24 or 30 sides.

For polyhedra, these values are given:

Polyhedra on the list: the following regular convex polyhedra with edges of length one: Platonic solids, four Archimedean solids (those with 14 faces or less).

The list was generated with this Python program.

Related pages on this site: properties of polygons, properties of polyhedra.

The numbers

There are 116 formulas on this page, with a total of 133 descriptions.

Approximate value Formula Descriptions
0.117851130√2 / 12tetrahedron – V
0.204124145√6 / 12tetrahedron – r
0.288675135√3 / 6triangle – r
0.353553391√2 / 4tetrahedron – ρ
0.408248290√6 / 6octahedron – r
0.433012702√3 / 4triangle – A
0.471404521√2 / 3octahedron – V
0.5000000001 / 2square – r
cube – r
octahedron – ρ
0.577350269√3 / 3triangle – R
0.612372436√6 / 4tetrahedron – R
truncated tetrahedron – r6
0.688190960√(25 + 10√5) / 10pentagon – r
0.707106781√2 / 2square – R
cube – ρ
octahedron – R
cuboctahedron – r4
0.755761314√3 × (3 + √5) / 12icosahedron – r
0.809016994(1 + √5) / 4icosahedron – ρ
0.816496581√6 / 3cuboctahedron – r3
0.850650808√(50 + 10√5) / 10pentagon – R
0.866025404√3 / 2triangle – h
hexagon – r
cube – R
cuboctahedron – ρ
0.951056516√(10 + 2√5) / 4icosahedron – R
1.000000001square – h
square – A
hexagon – R
cube – V
cuboctahedron – R
1.020620735√6 / 12truncated tetrahedron – r3
1.038260701 / (2 × tan(π / 7))heptagon – r
1.04719755π / 3triangle – internal angle in radians
1.060660173√2 / 4truncated tetrahedron – ρ
1.11351636√(250 + 110√5) / 20dodecahedron – r
1.152382441 / (2 × sin(π / 7))heptagon – R
1.17260394√22 / 4truncated tetrahedron – R
1.20710678(1 + √2) / 2octagon – r
truncated cube – r8
1.22474487√6 / 2truncated octahedron – r6
1.306562961 / √(2 − √2)octagon – R
1.30901699(3 + √5) / 4dodecahedron – ρ
1.373738711 / (2 × tan(π / 9))nonagon – r
1.40125854√3 × (1 + √5) / 4dodecahedron – R
1.41421356√2truncated octahedron – r4
1.461902201 / (2 × sin(π / 9))nonagon – R
1.500000003 / 2truncated octahedron – ρ
1.53884177√(5 + 2√5) / 2pentagon – h
decagon – r
1.57079633π / 2square – internal angle in radians
1.58113883√10 / 2truncated octahedron – R
1.61803399(1 + √5) / 2decagon – R
1.68252198√(51 + 36√2) / 6truncated cube – r3
1.70710678(2 + √2) / 2truncated cube – ρ
1.72047740√(25 + 10√5) / 4pentagon – A
1.73205081√3hexagon – h
tetrahedron – S
1.77882365√(7 + 4√2) / 2truncated cube – R
1.86602540(2 + √3) / 2dodecagon – r
1.884955593π / 5pentagon – internal angle in radians
1.93185165√(2 + √3)dodecagon – R
2.094395102π / 3hexagon – internal angle in radians
2.181694995 × (3 + √5) / 12icosahedron – V
2.19064313tan(3π / 7) / 2heptagon – h
2.243994755π / 7heptagon – internal angle in radians
2.35231505(√3 + √15 + √(10 + 2√5)) / 4pentadecagon – r
2.356194493π / 4octagon – internal angle in radians
2.357022605√2 / 3cuboctahedron – V
2.40486717(√3 + √(5 + 2√5)) / 2pentadecagon – R
2.414213561 + √2octagon – h
2.443460957π / 9nonagon – internal angle in radians
2.513274124π / 5decagon – internal angle in radians
2.51366975(1 + √2 + √(4 + 2√2)) / 2hexadecagon – r
2.562915451 / √(2 − √(2 + √2))hexadecagon – R
2.598076213√3 / 2hexagon – A
2.617993885π / 6dodecagon – internal angle in radians
2.7105759923√2 / 12truncated tetrahedron – V
2.7227136313π / 15pentadecagon – internal angle in radians
2.748893577π / 8hexadecagon – internal angle in radians
2.827433399π / 10icosagon – internal angles in radians
2.83564091tan(4π / 9) / 2nonagon – h
2.8797932711π / 12icositetragon – internal angles in radians
2.9321531414π / 15triacontagon – internal angles in radians
3.07768354√(5 + 2√5)decagon – h
3.15687576(1 + √5 + √(5 + 2√5)) / 2icosagon – r
3.196226612 / √(8 − 2√(10 + 2√5))icosagon – R
3.464101622√3octahedron – S
3.633912447 / (4 × tan(π / 7))heptagon – A
3.732050812 + √3dodecagon – h
3.79787706(2 + √2 + √3 + √6) / 2icositetragon – r
3.830648791 / √(2 − √(2 + √3))icositetragon – R
4.75718223(3√3 + √15 + √(50 + 22√5)) / 4pentadecagon – h
triacontagon – r
4.78338612(2 + √5 + √(15 + 6√5)) / 2triacontagon – R
4.828427122 + 2√2octagon – A
5.02733949(1 + √2 + √(4 + 2√2))hexadecagon – h
6.000000006cube – S
6.181824199 / (4 × tan(π / 9))nonagon – A
6.31375151(1 + √5 + √(5 + 2√5))icosagon – h
7.59575411(2 + √2 + √3 + √6)icositetragon – h
7.66311896(15 + 7√5) / 4dodecahedron – V
7.694208845√(5 + 2√5) / 2decagon – A
8.660254045√3icosahedron – S
9.464101626 + 2√3cuboctahedron – S
9.51436445(√15 + 3√3 + √(50 + 22√5)) / 2triacontagon – h
11.19615246 + 3√3dodecagon – A
11.31370858√2truncated octahedron – V
12.12435577√3truncated tetrahedron – S
13.59966337 × (3 + 2√2) / 3truncated cube – V
17.642362915 × (√3 + √15 + √(10 + 2√5)) / 8pentadecagon – A
20.10935804 × (1 + √2 + √(4 + 2√2))hexadecagon – A
20.64572883√(25 + 10√5)dodecahedron – S
26.78460976 × (1 + 2√3)truncated octahedron – S
31.56875765 × (1 + √5 + √(5 + 2√5))icosagon – A
32.43466442 × (6 + 6√2 + √3)truncated cube – S
45.57452476 × (2 + √2 + √3 + √6)icositetragon – A
60.000000060triangle – internal angle in degrees
71.357733415 × (√15 + 3√3 + √(50 + 22√5)) / 4triacontagon – A
90.000000090square – internal angle in degrees
108.000000108pentagon – internal angle in degrees
120.000000120hexagon – internal angle in degrees
128.571429900 / 7heptagon – internal angle in degrees
135.000000135octagon – internal angle in degrees
140.000000140nonagon – internal angle in degrees
144.000000144decagon – internal angle in degrees
150.000000150dodecagon – internal angle in degrees
156.000000156pentadecagon – internal angle in degrees
157.500000315 / 2hexadecagon – internal angle in degrees
162.000000162icosagon – internal angles in degrees
165.000000165icositetragon – internal angles in degrees
168.000000168triacontagon – internal angles in degrees