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Regular Polyhedron A solid, three-dimensional figure each face of which is a regular polygon with equal sides and equal angles. Every face has the same number of vertices, and the same number of faces meet at every vertex. An inscribed (inside) sphere touches the center of every face, and a circumscribed sphere (outside) touches every vertex. There are five and only five of these figures, also called the Platonic Solids: LET....................................................
	Number of vertices: v Number of edges: e Number of faces: f Edge: a Radius of circumscribed sphere: R Radius of inscribed sphere: r Surface area: S Volume: V Dihedral angle between faces: delta (in degrees)

Tetrahedron
	A three-dimensional figure with 4 equilateral triangle faces, 4 vertices, and 6 edges.
		v = 4, e = 6, f = 4 a = (2 sqrt[6]/3)R r = (1/3)R R = (sqrt[6]/4)a S = sqrt(3)a² V = (sqrt[2]/12)a³ delta = arccos(1/3) = 70^o 32' h = height or altitude h = (sqrt[6]/3)a

Cube
	A three-dimensional figure with 6 square faces, 8 vertices, and 12 edges.
		v = 8, e = 12, f = 6 a = (2 sqrt[3]/3)R r = (sqrt[3]/3)R R = (sqrt[3]/2)a r = (1/2)a S = 6 a² V = a³ delta = arccos(0) = 90^o

Octahedron
	A three-dimensional figure with 8 equilateral triangle faces, 6 vertices, and 12 edges.
		v = 6, e = 12, f = 8 a = sqrt(2)R r = (sqrt[3]/3)R R = (sqrt[2]/2)a r = (sqrt[6]/6)a S = 2 sqrt(3)a² V = (sqrt[2]/3)a³ delta = arccos(-1/3) = 109^o 28'

Dodecahedron
	A three-dimensional figure with 12 regular pentagon faces, 20 vertices, and 30 edges.
		v = 20, e = 30, f = 12 a = ([sqrt(5)-1]sqrt[3]/3)R r = (sqrt[75+30 sqrt(5)]/15)R R = (sqrt[3][1+sqrt[5])/4)a r = (sqrt[250+110 sqrt(5)]/20)a S = 3 sqrt(25+10 sqrt[5])a² V = ([15+7 sqrt(5)]/4)a³ delta = arccos(-sqrt[5]/5) = 116^o 34'

Icosahedron
	A three-dimensional figure with 20 equilateral triangle faces, 12 vertices, and 30 edges.
		v = 12, e = 30, f = 20 a = (sqrt[50-10 sqrt(5)]/5)R r = (sqrt[75+30 sqrt(5)]/15)R R = (sqrt[10+2 sqrt(5)]/4)a r = (sqrt[42+18 sqrt(5)]/12)a S = 5 sqrt(3)a² V = (5[3+sqrt(5)]/12)a³ delta = arccos(-sqrt[5]/3) = 138^o 11'

NOW ABOUT CYLINDERS................!!!!!!!!!!!!!!!!!!!!

Circular Cylinder A cylinder whose bases are circles. The line connecting the centers of the bases is called the axis.
Height: h Area of base: B Length of lateral edge: l Area of right section: A Perimeter of right section: P Lateral surface area: S Total surface area: T Volume: V S = lP T = lP + 2B V = hB = lA
Right Circular Cylinder A circular cylinder in which the axis is perpendicular to the bases. (If the axis of a circular cylinder is not perpendicular to the bases, it is called an oblique circular cylinder.)
Height: h Radius of base: r Lateral surface area: S Total surface area: T Volume: V S = 2 Pi rh T = 2 Pi r(r+h) V = Pi r²h A = B = Pi r² P = 2 Pi r l = h