Three-dimensional geometric shapes

Platonic solids

Keywords: Platonic solids, also called the regular solids or regular polyhedra

Name	Tetrahedron	Hexahedron (Cube)	Octahedron	Dodecahedron	Icosahedron
Picture
Net
Faces	4	6	8	12	20
Face type	equilateral triangle	square	equilateral triangle	equilateral pentagon	equilateral triangle
Duality	tetrahedron	octahedron	hexahedron	icosahedron	dodecahedron
Edges	6	12	12	30	30
Vertices	4	8	6	20	12
Edges from vertex	3	3	4	3	5
Number of diagonals	0	4	3	100	36
Dihedral angle	≈70°31'43,61''	90°	≈109°28'16,39''	≈116°33'55,84''	≈138°11'22,87''
Surface area	$\sqrt{3} a^{2}$	$6 a^{2}$	$2 \sqrt{3} a^{2}$	$3 \sqrt{25 + 10 \sqrt{5}} a^{2}$	$5 \sqrt{3} a^{2}$
Volume	$\frac{\sqrt{2} a^{3}}{12}$	$a^{3}$	$\frac{\sqrt{2} a^{3}}{3}$	$\frac{(15 + 7 \sqrt{5}) a^{3}}{4}$	$\frac{(15 + 5 \sqrt{5}) a^{3}}{12}$
Circumradiu	$\frac{\sqrt{6} a}{4}$	$\frac{\sqrt{3} a}{2}$	$\frac{\sqrt{3} a}{2}$	$\frac{\sqrt{3} (1 + \sqrt{5}) a}{4}$	$\frac{\sqrt{10 + 2 \sqrt{5}} a}{4}$
Inradiu	$\frac{\sqrt{6} a}{12}$	$\frac{a}{2}$	$\frac{\sqrt{6} a}{6}$	$\frac{1}{2} \sqrt{\frac{25 + 11 \sqrt{5}}{10}} a$	$\frac{\sqrt{42 + 18 \sqrt{5}}}{12} a$
Midradius	$\frac{\sqrt{2} a}{4}$	$\frac{\sqrt{2} a}{2}$	$\frac{a}{2}$	$\frac{(\sqrt{5} + 3) a}{4}$	$\frac{(1 + \sqrt{5}) a}{4}$

Trigonometry
Law of Sines
Law of Cosines
Trygonometry Identities of same angle
Identities for the sum and difference of two angles
Trigonometric identities of double angles
Trigonometric identities of half angles
Sum and difference of trigonometric functions
Trigonometry of right triangles
Two-dimensional geometric shapes
Basic Proportionality Theorem
Triangle
Special Triangles - Right Triangle, Equilateral Triangles, Isosceles Triangles
Quadrilateral
Squere
Rectangle
Rhombus
Parallelogram
Kite
Trapezoid
Circle
Circular sector
Circular segment
Three-dimensional geometric shapes
Cube
Sphere
Cone
Prism
Pyramide
Platonic solids
Cylinder
Vectors
Scalar Product of Vectors
Vector Product of Vectors
Analytical Geometry 2D
Distance between two points
Circle
Linear equation
Ellipse