Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

So much of geometry takes place in a flattened world alien to our own three-dimensional existence. Polyhedrons (or polyhedra), on the other hand, are familiar objects; they are solids with flat faces, and they are all around us.

## Polyhedron Definition

You have some experience with polygons, "many-angled" shapes that exist in two dimensions. Polyhedrons are the three-dimensional relatives of polygons. The word "polyhedron" means "many seated" or "many based," since the faces of three-dimensional shapes are their bases.

The plural of polyhedron can be either polyhedra or polyhedrons. To be a polyhedron, the three-dimensional shape must have width, depth and length, and every face must be composed of polygons. That makes every polyhedron sharp-edged, with clean, straight lines.

## Polyhedron Characteristics

Polyhedra have three identifying properties, or traits that make them uniquely polyhedrons. They must be:

• Three-dimensional
• Solid
• Composed of only polygons

A favorite stuffed animal fulfills the first two conditions, as do most of the everyday items you own (TV, wristwatch, laptop, sporting goods, and so on). Yet the third identifying property really narrows the field, because no curves of any kind are allowed. Your stuffed animals are not polyhedra. Your laptop and wristwatch probably have curves in them.

Most sporting goods are not polyhedra. Consider a softball. It is made of curved surfaces, so it is not a polyhedron. Yet a soccer ball is a polyhedron; its "curves" are made from 12 pentagons and 20 hexagons. A soccer ball is a truncated icosahedron, one of the many types of polyhedra.

To identify a polyhedron, check its edges. They should be sharp and straight. Check out tomorrow morning's box of cereal. It has 12 straight edges joining six flat faces. It is a rectangular prism, a kind of polyhedron.

## Polyhedron Examples

Notice no face has a curve, all faces are polygons, and the three-dimensional, solid shape uses the same shape over and over.

[insert animation of an octahedron or other simple polyhedron rotating to reveal all its faces]

### Real World Polyhedra

Since we are three-dimensional creatures, humans live in and among polyhedra. Consider all these:

• The Great Pyramid and all the lesser pyramids
• Above-ground pools
• Bricks
• Concrete blocks
• Dice
• Houses (if built with only straight walls and roof lines; no curves)
• Prisms (triangular, rectangular, hexagonal and more)
• Office desks
• Shipping cartons
• Concrete pavers and landscaping stones

Many polyhedra follow the same basic structure: a polygon is repeated on itself to grow into the three-dimensional world. Imagine an equilateral triangle and its three identical siblings all joined along their sides. You get a tetrahedron, or four-sided, triangular pyramid.

Imagine a square and five of its siblings all joining along their sides; you get a cube. A rectangle can be expanded to form a rectangular prism (like a brick, cereal box, or shipping carton); a hexagon can gain thickness to become a hexagonal prism, making an excellent paving stone for gardens, paths and driveways.

Many above-ground pools are very large polyhedrons. Your next board game that uses dice will mean you will be playing with polyhedra.

We usually live inside polyhedra, often work in polyhedrons, and, as mentioned before, kick truncated icosahedrons around on soccer fields.

## Lesson Summary

Now that you have enjoyed and studied this video lesson, you are able to explain what a polyhedron is, recall and state the three identifying properties of polyhedra, and associate the mathematical models of polyhedrons with their real-world counterparts, like soccer balls and pyramids.

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