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.

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 degrees and radians
number of sides (n in n-gon)

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
number of vertices, edges and faces

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 192 formulas on this page, with a total of 264 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
0.5	1 / 2	square – inradius
		cube – inradius
		octahedron – midradius
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	1	all polygons on this list – side
		square – height
		square – area
		hexagon – circumradius
		circle – radius
		all polyhedra on this list – edge
		cube – volume
		cuboctahedron – circumradius
		sphere – radius
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
1.27427369	(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
1.5	3 / 2	truncated octahedron – midradius
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
1.58113883	√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
1.91421356	(1 + 2√2) / 2	truncated cuboctahedron – inradius w.r.t. octagons
1.93185165	√(2 + √3)	dodecagon – circumradius
2	2	circle – diameter
2	2	sphere – diameter
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
2.41421356	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	3	triangle – number of sides
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	4	square – number of sides
		tetrahedron – number of faces
		tetrahedron – number of vertices
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	5	pentagon – number of sides
5.02733949	(1 + √2 + √(4 + 2√2))	hexadecagon – height
6	6	hexagon – number of sides
		tetrahedron – number of edges
		cube – number of faces
		cube – surface area
		octahedron – number of vertices
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	7	heptagon – number of sides
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	8	octagon – number of sides
		cube – number of vertices
		octahedron – number of faces
		truncated tetrahedron – number of faces
8.66025404	5√3	icosahedron – surface area
8.71404521	(12 + 10√2) / 3	rhombicuboctahedron – volume
9	9	nonagon – number of sides
9.46410162	6 + 2√3	cuboctahedron – surface area
9.51436445	(√15 + 3√3 + √(50 + 22√5)) / 2	triacontagon – height
10	10	decagon – number of sides
11.1961524	6 + 3√3	dodecagon – area
11.3137085	8√2	truncated octahedron – volume
12	12	dodecagon – number of sides
		cube – number of edges
		octahedron – number of edges
		dodecahedron – number of faces
		icosahedron – number of vertices
		truncated tetrahedron – number of vertices
		cuboctahedron – number of vertices
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
14	14	cuboctahedron – number of faces
		truncated cube – number of faces
		truncated octahedron – number of faces
15	15	pentadecagon – number of sides
16	16	hexadecagon – number of sides
17.6423629	15 × (√3 + √15 + √(10 + 2√5)) / 8	pentadecagon – area
18	18	truncated tetrahedron – number of edges
19.8564065	6 + 8√3	snub cube – surface area
20	20	icosagon – number of sides
		dodecahedron – number of vertices
		icosahedron – number of faces
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
24	24	icositetragon – number of sides
		cuboctahedron – number of edges
		truncated cube – number of vertices
		truncated octahedron – number of vertices
		rhombicuboctahedron – number of vertices
		snub cube – number of vertices
26	26	rhombicuboctahedron – number of faces
26	26	truncated cuboctahedron – number of faces
26.7846097	6 × (1 + 2√3)	truncated octahedron – surface area
29.3059828	5√3 + 3√(25 + 10√5)	icosidodecahedron – surface area
30	30	triacontagon – number of sides
		dodecahedron – number of edges
		icosahedron – number of edges
		icosidodecahedron – number of vertices
31.5687576	5 × (1 + √5 + √(5 + 2√5))	icosagon – area
32	32	icosidodecahedron – number of faces
		truncated dodecahedron – number of faces
		truncated icosahedron – number of faces
32.4346644	2 × (6 + 6√2 + √3)	truncated cube – surface area
36	36	truncated cube – number of edges
36	36	truncated octahedron – number of edges
38	38	snub cube – number of faces
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
48	48	rhombicuboctahedron – number of edges
48	48	truncated cuboctahedron – number of vertices
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
60	60	triangle – internal angle in degrees
		snub cube – number of edges
		icosidodecahedron – number of edges
		truncated dodecahedron – number of vertices
		truncated icosahedron – number of vertices
		rhombicosidodecahedron – number of vertices
		snub dodecahedron – number of vertices
61.7551724	12 × (2 + √2 + √3)	truncated cuboctahedron – surface area
62	62	rhombicosidodecahedron – number of faces
62	62	truncated icosidodecahedron – number of faces
71.3577334	15 × (√15 + 3√3 + √(50 + 22√5)) / 4	triacontagon – area
72	72	truncated cuboctahedron – number of edges
72.6072530	30√3 + 3√(25 + 10√5)	truncated icosahedron – surface area
85.0396646	5 × (99 + 47√5) / 12	truncated dodecahedron – volume
90	90	square – internal angle in degrees
		truncated dodecahedron – number of edges
		truncated icosahedron – number of edges
92	92	snub dodecahedron – number of faces
100.990760	5 × (√3 + 6√(5 + 2√5))	truncated dodecahedron – surface area
108	108	pentagon – internal angle in degrees
120	120	hexagon – internal angle in degrees
		rhombicosidodecahedron – number of edges
		truncated icosidodecahedron – number of vertices
128.571429	900 / 7	heptagon – internal angle in degrees
135	135	octagon – internal angle in degrees
140	140	nonagon – internal angle in degrees
144	144	decagon – internal angle in degrees
150	150	dodecagon – internal angle in degrees
150	150	snub dodecahedron – number of edges
156	156	pentadecagon – internal angle in degrees
157.5	315 / 2	hexadecagon – internal angle in degrees
162	162	icosagon – internal angles in degrees
165	165	icositetragon – internal angles in degrees
168	168	triacontagon – internal angles in degrees
174.292030	30 × (1 + √3 + √(5 + 2√5))	truncated icosidodecahedron – surface area
180	180	truncated icosidodecahedron – number of edges
206.803399	95 + 50√5	truncated icosidodecahedron – volume
360	360	degrees per full turn