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These are NOT polyhedron:

These are polyhedron:

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Crystallographicpolyhedra

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-three dimensional;-a closed figure;-made up of flat surfaces which are polygons.

What is a polyhedron?A polyhedron is a three-dimensional solid whose faces are polygons joined at their edges. The word polyhedron is derived from the Greek poly (many) and the Indo-European hedron (seat).

All of the faces (sides) of a polyhedron are polygons.

The line segments where the faces intersect are called edges.

The point where more than two faces intersect is called a vertex.

edges vertex

face

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A polyhedron is said to be regular if its faces are made up of regular polygons.

There are only five regular polyhedra called the Platonic solids.

Crystals are real world examples of polyhedra. The salt you sprinkle on your food is a crystal in the

shape of a cube.

In mathematics Plato's name is attached to the Platonic solids. In the Timaeus there is a mathematical construction of the elements (earth, fire, air, and water), in which the cube, tetrahedron, octahedron, and icosahedron are given as the shapes of the atoms of earth, fire, air, and water. The fifth Platonic solid, the dodecahedron, is Plato's model for the whole universe.

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-is a polyhedron that has all faces except one intersecting at one point;-has one polygon base;The sides that are not the base and intersect in a single point are triangles.-is named by the shape of its base.

Pentagonal pyramid

Triangular pyramid

Hexagonal pyramid

-is a polyhedron with two congruent bases that are in parallel planes.The faces that are not bases are parallelograms, called lateral faces.-is named by the shape of its bases.

A B

C

D

E

F

A solid with a pair of circular bases is called a cylinder. Is a cylinder a polyhedron?

A cone has a circular base and a vertex. Is a cone a polyhedron?

letter Name of polyhedron

Number of faces

Number of edges

Number of vertices

A

B

C

D

E

F

G

H

Complete the table below.

letter Name of polyhedron

Number of faces

Number of edges

Number of vertices

A Triangular prism 5 9 6

B Triangular pyramid 4 6 4

C Square prism 6 12 8

D Pentagonal prism 7 15 10

E Hexagonal pyramid 7 12 7

F Octagonal prism 10 24 16

G Rectangular pyramid 5 8 5

H Septagonal prism 9 21 14n = # sides

on basePyramid / prism n + 1 / n + 2 2n / 3n n + 1 / 2n

Completed table:

What is the relationship between the number of vertices, faces and edges in a polyhedron?

Click here for proofsof Euler’s formula

For any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges.

V + F = 2 + E

This is called Euler’s formula.

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