List of numbers related to geometry

Introduction

If you come across an irrational number and think it might be related to geometry, you may find the corresponding formula here.

For polygons, these values are given:

circumradius R
inradius r
height h (equals 2r for even n-gons and R+r for odd n-gons)
area A
internal angle in radians

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:

circumradius R
midradius ρ (small rho)
inradius for all faces r or inradius for n-gonal faces r_n
surface area S
volume V
dihedral angle (Platonic solids only)

Polyhedra on the list: Platonic and Archimedean solids with edges of length one.

Other shapes on the list: circle and sphere, both with a radius of one.

The list was generated with this Python program.

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

The numbers

There are 156 formulas on this page, with a total of 174 descriptions.

Approximate value	Formula	Descriptions
0.0174532925	`π` / 180	radians per degree
0.117851130	√2 / 12	tetrahedron – volume
0.204124145	√6 / 12	tetrahedron – inradius
0.288675135	√3 / 6	triangle – inradius
0.353553391	√2 / 4	tetrahedron – midradius
0.408248290	√6 / 6	octahedron – inradius
0.433012702	√3 / 4	triangle – area
0.471404521	√2 / 3	octahedron – volume
0.577350269	√3 / 3	triangle – circumradius
0.612372436	√6 / 4	tetrahedron – circumradius
0.612372436	√6 / 4	truncated tetrahedron – inradius w.r.t. hexagons
0.688190960	√(25 + 10√5) / 10	pentagon – inradius
0.707106781	√2 / 2	square – circumradius
		cube – midradius
		octahedron – circumradius
		cuboctahedron – inradius w.r.t. squares
0.755761314	√3 × (3 + √5) / 12	icosahedron – inradius
0.809016994	(1 + √5) / 4	icosahedron – midradius
0.816496581	√6 / 3	cuboctahedron – inradius w.r.t. triangles
0.850650808	√(50 + 10√5) / 10	pentagon – circumradius
0.866025404	√3 / 2	triangle – height
		hexagon – inradius
		cube – circumradius
		cuboctahedron – midradius
0.951056516	√(10 + 2√5) / 4	icosahedron – circumradius
1.02062073	5√6 / 12	truncated tetrahedron – inradius w.r.t. triangles
1.03826070	1 / (2 × tan(`π` / 7))	heptagon – inradius
1.04719755	`π` / 3	triangle – internal angle in radians
1.06066017	3√2 / 4	truncated tetrahedron – midradius
1.11351636	√(250 + 110√5) / 20	dodecahedron – inradius
1.15238244	1 / (2 × sin(`π` / 7))	heptagon – circumradius
1.17260394	√22 / 4	truncated tetrahedron – circumradius
1.20710678	(1 + √2) / 2	octagon – inradius
		truncated cube – inradius w.r.t. octagons
		rhombicuboctahedron – inradius w.r.t. squares
1.22474487	√6 / 2	truncated octahedron – inradius w.r.t. hexagons
1.23095942	acos(1 / 3)	tetrahedron – dihedral angle in radians
1.27427369	(3√3 + √6) / 6	rhombicuboctahedron – inradius w.r.t. triangles
1.30656296	1 / √(2 − √2)	octagon – circumradius
1.30656296	1 / √(2 − √2)	rhombicuboctahedron – midradius
1.30901699	(3 + √5) / 4	dodecahedron – midradius
1.37373871	1 / (2 × tan(`π` / 9))	nonagon – inradius
1.37638192	√(25 + 10√5) / 5	icosidodecahedron – inradius w.r.t. pentagons
1.39896633	√(5 + 2√2) / 2	rhombicuboctahedron – circumradius
1.40125854	√3 × (1 + √5) / 4	dodecahedron – circumradius
1.41421356	√2	truncated octahedron – inradius w.r.t. squares
1.46190220	1 / (2 × sin(`π` / 9))	nonagon – circumradius
1.51152263	√(42 + 18√5) / 6	icosidodecahedron – inradius w.r.t. triangles
1.53884177	√(5 + 2√5) / 2	pentagon – height
		decagon – inradius
		icosidodecahedron – midradius
1.57079633	`π` / 2	square – internal angle in radians
1.57079633	`π` / 2	cube – dihedral angle in radians
1.58113883	√10 / 2	truncated octahedron – circumradius
1.61803399	(1 + √5) / 2	decagon – circumradius
		icosidodecahedron – circumradius
		the golden ratio
1.68252198	√(51 + 36√2) / 6	truncated cube – inradius w.r.t. triangles
1.70710678	(2 + √2) / 2	truncated cube – midradius
1.72047740	√(25 + 10√5) / 4	pentagon – area
1.73205081	√3	hexagon – height
1.73205081	√3	tetrahedron – surface area
1.77882365	√(7 + 4√2) / 2	truncated cube – circumradius
1.83928676	(1 + ∛(19 + 3√33) + ∛(19 − 3√33)) / 3	tribonacci constant
1.86602540	(2 + √3) / 2	dodecagon – inradius
1.88495559	3`π` / 5	pentagon – internal angle in radians
1.91063324	acos(−1 / 3)	octahedron – dihedral angle in radians
1.91421356	(1 + 2√2) / 2	truncated cuboctahedron – inradius w.r.t. octagons
1.93185165	√(2 + √3)	dodecagon – circumradius
2.03444394	acos(−√5 / 5)	dodecahedron – dihedral angle in radians
2.06457288	3√(25 + 10√5) / 10	rhombicosidodecahedron – inradius w.r.t. pentagons
2.09077028	(√6 + √3) / 2	truncated cuboctahedron – inradius w.r.t. hexagons
2.09439510	2`π` / 3	hexagon – internal angle in radians
2.11803399	(2 + √5) / 2	rhombicosidodecahedron – inradius w.r.t. squares
2.15701985	(3√3 + 2√15) / 6	rhombicosidodecahedron – inradius w.r.t. triangles
2.17625090	√(10 + 4√5) / 2	rhombicosidodecahedron – midradius
2.18169499	5 × (3 + √5) / 12	icosahedron – volume
2.19064313	tan(3`π` / 7) / 2	heptagon – height
2.20710678	(3 + √2) / 2	truncated cuboctahedron – inradius w.r.t. squares
2.23295051	√(11 + 4√5) / 2	rhombicosidodecahedron – circumradius
2.24399475	5`π` / 7	heptagon – internal angle in radians
2.26303344	√(12 + 6√2) / 2	truncated cuboctahedron – midradius
2.26728394	√(42 + 18√5) / 4	truncated icosahedron – inradius w.r.t. hexagons
2.31761091	√(13 + 6√2) / 2	truncated cuboctahedron – circumradius
2.32743844	√(1250 + 410√5) / 20	truncated icosahedron – inradius w.r.t. pentagons
2.35231505	(√3 + √15 + √(10 + 2√5)) / 4	pentadecagon – inradius
2.35619449	3`π` / 4	octagon – internal angle in radians
2.35702260	5√2 / 3	cuboctahedron – volume
2.40486717	(√3 + √(5 + 2√5)) / 2	pentadecagon – circumradius
2.41186500	acos(−√5 / 3)	icosahedron – dihedral angle in radians
2.41421356	1 + √2	octagon – height
2.42705098	(3 + 3√5) / 4	truncated icosahedron – midradius
2.44346095	7`π` / 9	nonagon – internal angle in radians
2.47801866	√(58 + 18√5) / 4	truncated icosahedron – circumradius
2.48989828	√(50 + 22√5) / 4	truncated dodecahedron – inradius w.r.t. decagons
2.51327412	4`π` / 5	decagon – internal angle in radians
2.51366975	(1 + √2 + √(4 + 2√2)) / 2	hexadecagon – inradius
2.56291545	1 / √(2 − √(2 + √2))	hexadecagon – circumradius
2.59807621	3√3 / 2	hexagon – area
2.61799388	5`π` / 6	dodecagon – internal angle in radians
2.71057599	23√2 / 12	truncated tetrahedron – volume
2.72271363	13`π` / 15	pentadecagon – internal angle in radians
2.74889357	7`π` / 8	hexadecagon – internal angle in radians
2.82743339	9`π` / 10	icosagon – internal angles in radians
2.83564091	tan(4`π` / 9) / 2	nonagon – height
2.87979327	11`π` / 12	icositetragon – internal angles in radians
2.91278117	√3 × (9 + 5√5) / 12	truncated dodecahedron – inradius w.r.t. triangles
2.92705098	(5 + 3√5) / 4	truncated dodecahedron – midradius
2.93215314	14`π` / 15	triacontagon – internal angles in radians
2.96944902	√(74 + 30√5) / 4	truncated dodecahedron – circumradius
3.07768354	√(5 + 2√5)	decagon – height
3.14159265	`π`	circle – area
3.15687576	(1 + √5 + √(5 + 2√5)) / 2	icosagon – inradius
3.19622661	2 / √(8 − 2√(10 + 2√5))	icosagon – circumradius
3.44095480	√(25 + 10√5) / 2	truncated icosidodecahedron – inradius w.r.t. decagons
3.46410162	2√3	octahedron – surface area
3.63391244	7 / (4 × tan(`π` / 7))	heptagon – area
3.66854248	(2√3 + √15) / 2	truncated icosidodecahedron – inradius w.r.t. hexagons
3.73205081	2 + √3	dodecagon – height
3.73606798	(3 + 2√5) / 2	truncated icosidodecahedron – inradius w.r.t. squares
3.76937713	√(30 + 12√5) / 2	truncated icosidodecahedron – midradius
3.79787706	(2 + √2 + √3 + √6) / 2	icositetragon – inradius
3.80239450	√(31 + 12√5) / 2	truncated icosidodecahedron – circumradius
3.83064879	1 / √(2 − √(2 + √3))	icositetragon – circumradius
4.18879020	4`π` / 3	sphere – volume
4.75718223	(3√3 + √15 + √(50 + 22√5)) / 4	pentadecagon – height
4.75718223	(3√3 + √15 + √(50 + 22√5)) / 4	triacontagon – inradius
4.78338612	(2 + √5 + √(15 + 6√5)) / 2	triacontagon – circumradius
4.82842712	2 + 2√2	octagon – area
5.02733949	(1 + √2 + √(4 + 2√2))	hexadecagon – height
6.18182419	9 / (4 × tan(`π` / 9))	nonagon – area
6.28318531	2`π`	circle – circumference
6.28318531	2`π`	radians per full turn
6.31375151	(1 + √5 + √(5 + 2√5))	icosagon – height
7.59575411	(2 + √2 + √3 + √6)	icositetragon – height
7.66311896	(15 + 7√5) / 4	dodecahedron – volume
7.69420884	5√(5 + 2√5) / 2	decagon – area
8.66025404	5√3	icosahedron – surface area
8.71404521	(12 + 10√2) / 3	rhombicuboctahedron – volume
9.46410162	6 + 2√3	cuboctahedron – surface area
9.51436445	(√15 + 3√3 + √(50 + 22√5)) / 2	triacontagon – height
11.1961524	6 + 3√3	dodecagon – area
11.3137085	8√2	truncated octahedron – volume
12.1243557	7√3	truncated tetrahedron – surface area
12.5663706	4`π`	sphere – surface area
13.5996633	7 × (3 + 2√2) / 3	truncated cube – volume
13.8355259	(45 + 17√5) / 6	icosidodecahedron – volume
17.6423629	15 × (√3 + √15 + √(10 + 2√5)) / 8	pentadecagon – area
19.8564065	6 + 8√3	snub cube – surface area
20.1093580	4 × (1 + √2 + √(4 + 2√2))	hexadecagon – area
20.6457288	3√(25 + 10√5)	dodecahedron – surface area
21.4641016	18 + 2√3	rhombicuboctahedron – surface area
26.7846097	6 × (1 + 2√3)	truncated octahedron – surface area
29.3059828	5√3 + 3√(25 + 10√5)	icosidodecahedron – surface area
31.5687576	5 × (1 + √5 + √(5 + 2√5))	icosagon – area
32.4346644	2 × (6 + 6√2 + √3)	truncated cube – surface area
41.6153238	(60 + 29√5) / 3	rhombicosidodecahedron – volume
41.7989899	22 + 14√2	truncated cuboctahedron – volume
45.5745247	6 × (2 + √2 + √3 + √6)	icositetragon – area
55.2867450	20√3 + 3√(25 + 10√5)	snub dodecahedron – surface area
55.2877308	(125 + 43√5) / 4	truncated icosahedron – volume
57.2957795	180 / `π`	degrees per radian
59.3059828	30 + 5√3 + 3√(25 + 10√5)	rhombicosidodecahedron – surface area
61.7551724	12 × (2 + √2 + √3)	truncated cuboctahedron – surface area
70.5287794	degrees(acos(1 / 3))	tetrahedron – dihedral angle in degrees
71.3577334	15 × (√15 + 3√3 + √(50 + 22√5)) / 4	triacontagon – area
72.6072530	30√3 + 3√(25 + 10√5)	truncated icosahedron – surface area
85.0396646	5 × (99 + 47√5) / 12	truncated dodecahedron – volume
100.990760	5 × (√3 + 6√(5 + 2√5))	truncated dodecahedron – surface area
109.471221	degrees(acos(−1 / 3))	octahedron – dihedral angle in degrees
116.565051	degrees(acos(−√5 / 5))	dodecahedron – dihedral angle in degrees
138.189685	degrees(acos(−√5 / 3))	icosahedron – dihedral angle in degrees
174.292030	30 × (1 + √3 + √(5 + 2√5))	truncated icosidodecahedron – surface area
206.803399	95 + 50√5	truncated icosidodecahedron – volume