12.1 Exploring solids
Polyhedron - is a solid that is bounded by polygons called faces that enclose a single region of space.
An edge of a polyhedron is a line segment formed by the intersection of two faces.
A vertex of a polyhedron is a point where three of more edges meet.
Plural = polyhedra
Example:
Regular Polyhedron - all faces are congruent and all the faces are regular polygons (polygons that have all angles congruent and all sides congruent).
Convex Polyhedron - any two points on its surface can be connected by a segment that lies entirely inside or on the polyhedron.
Cross-section - intersection of the plane and the solid. pg. 720
Platonic Solids -
Regular Tetrahedron (4 faces, 4 vertices, 6 edges) - associated with the element of fire.
Cube (6 faces, 8 vertices, 12 edges) -
Regular Octahedron (8 faces, 6 vertices, 12 edges) - associated with the element of air.
Regular Dodecahedron (12 faces)
Regular Icosahedron (20 faces, 12 vertices, 30 edges) - associated with the element of water. A soccer ball’s pattern is a truncated icosahedron.
Euler’s Theorem:
The number of faces (F), vertices (V), and edges (E), of a polyhedron are related by the following formula:
F + V = E + 2
Example: a polyhedron has 20 faces; all triangle
Since a triangle has 3 sides and each of the triangles are going to share an edge, you have
20 + V = ((3)(20))/2 + 2
20 + V = (60)/2 + 2
20 + V = 30 + 2
20 + V = 32
V = 12
Homework: #40 pg. 723 #10-30 even, 32-35, 54, 55, 60-62, 69
