Tessellations - by popular demand! (7/00) (top of page)

 The Five Regular Polyhedra ("Platonic Solids")
         
Tetrahedron
< 3, 3, 3 >
4 triangles
Octahedron
< 3, 3, 3, 3 >
8 triangles
Hexahedron (cube)
< 4, 4, 4 >
6 squares
Icosahedron
< 3, 3, 3, 3, 3 >
20 triangles
Dodecahedron
< 5, 5, 5 >
12 pentagons

You can make your own regular and semi-regular polyhedra by cutting out dozens of cardboard polygons and using scotch tape for the edges. I used to sell these things as 'geometric art' when I was a kid. "Get your truncated icosahedron over here, one dollar!" (What sport uses this shape?)
 
 The Thirteen Semi-Regular Polyhedra ("Archimedean Solids")
Graphics produced with Mathematica
         
Truncated
Tetrahedron
< 3, 6, 6 >
4 triangles
4 hexagons
Truncated
Cube
< 3, 3, 3, 3 >
8 triangles
6 octagons
Cubocta-
hedron
< 4, 4, 4 >
8 triangles
6 squares
Truncated
Octahedron
< 5, 5, 5 >
6 squares
8 hexagons
Snub
Cube
< 3, 3, 3, 3, 4 >
32 triangles
6 squares
   
 It's a fun puzzle to
figure out, for each
solid, how many:
.vertices (V) (corners)
.edges (E) (edges) and
.faces (F) (polygons).
   
Icosi-
dodecahedron
< 3, 5, 3, 5 >
20 triangles
12 pentagons
Truncated
Cuboctahedron
< 4, 6, 8 >
12 sqrs, 8 hexs,
6 octagons
Rhombicosa-
cuboctahedron
< 3, 4, 4, 4 >
8 triangles
18 squares
Truncated
Dodecahedron
< 3, 10, 10 >
20 triangles
12 decagons
   
Make up variables
for the number of
each polygon.
Follow Euler's rule
V - E + F = 2.
Then solve for
V, E, F ; I dare you!
   
Truncated
Icosahedron
< 5, 6, 6 >
12 pentagons
20 hexagons
Truncated Icosi-
dodecahedron
< 4, 6, 10 >
30 sq, 20 hex,
12 decagons
Snub
Dodecahedron
< 3, 3, 3, 3, 5 >
80 triangles,
12 pentagons
Rhombicosa-
dodecahedron
< 3, 4, 5, 4 >
20 tri, 30 sq,
12 pentagons
 

Perimeter, Area, and Volume Formulas (or is it 'formulae') (3/01) (top of page)
1. The perimeter of a polygon (or any closed curve) is the distance around.
2. The area of a simple, closed, planar curve is the amount of space inside.
3. The volume of a solid 3D shape is the amount of space displaced by it.
 
There are plenty of good formulas for figuring these out: the answers have one,
two, or three dimensions; linear units, square units, or cubic units.
 
1. Perimeter formulas: P = 4s (square) , P = 2L + 2W (rectangle) , P = a + b + c (triangle) ,
. . . P = C = 2 pi r = pi D (circle) Also c =(a^2 + b^2) (right triangle; see trig page)
         
2. Area formulas: A = s^2 (square) , A = LW (rectangle) , A = (1/2) b h (triangle) ,
. . . A = [s(s - a)(s - b)(s - c)] (triangle; s = (a+b+c)/2) , A = b h (trapezoid) ,
. . . A = pi r^2 (circle)
         
3. Volume formulas: V = s^3 (cube) , V = LWH (rectang. box) , V = A h (any cylinder) ,
. . . (A = area of base) , V = (1/3) A h (any pyramid or cone) , V = (4/3) pi r^3 (sphere)..
 
** Pictures coming soon for the formula section! **

 
 
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