13.1: Polyhedra and Other Solid Shapes

names the component parts of solid shapes

(1) points, (2)edges, (3) flat outer surfaces (4) curved outer surfaces

describe cylinders

1) draw a closed curve on paper (such as a circle or an oval) 2) cut it out, make a copy of it, and lay the two copies on a table, one on top of the other, so that they match 3)take the top copy and move it up without twisting, away from the bottom copy, keeping the two copies parallel 4) now imagine paper or some other kind of material connecting the two curves in such a way that every line between corresponding points on the curves lies on this paper or material

describe a cone

1) start with a closed curve in a plane and a separate point that does not lie in that plane 2) imagine paper or some other kind of material that joins the point to the curve in such a way that all the lines that connect the point to the curve lie on that paper (or other material) that paper or other material together with original curve form a cone

describe a pyramid

1) start with any polygon and separate a single point, called an apex , that does not lie in the plane of the polygon 2) use additional polygons to connect the apex to the original polygon. this should be done in such a way that all the lines that connect the apex to the polygon lie on the side of these additional polygons these new polygons, together with the original polygon, form a pyramid

describe how right prisms are formed

1) take two paper copies of any polygon and lay both flat on a table, one on top of the other so that they match up. 2) move the top polygon straight up above the bottom one 3) if vertical rectangular faces are now placed so as to connect corresponding sides of the two polygons, then the shape formed this way is a right prism

describe how to form all prisms, not just right prisms

1) take two paper copies of any polygon and lay both flat on a table, one on top of the other so that they match up. 2) move the top polygon, without twisting, away from the bottom polygon, keeping the two polygons parallel. 3) this time the top polygon does not need to go straight up over the bottom polygon. 4) if faces are now placed so as to connect corresponding sides of the two polygons, then the shape formed this way is a prism this time the faces will be parallelograms

ex: icosahedron means

20 bases; made of 20 triangular bases

what is a polyhedron (polyhedra)

a closed, connected shape in space whose outer surfaces consist of polygons such as triangles, squares, or pentagons...

what's an oblique cone

a cone that is not a right cone

what's a vertex or corner of the polyhedron

a corner point where several faces come together plural of vertex is vertices

what's a right prism

a polyhedron that can thought of as going straight up over a polygon

what's an oblique prism

a prism that not a right prism

what's an oblique pyramid

a pyramid that is not a right pyramid

what's the difference between a rectangle and a rectangular prism

a rectangle is a two dimensional shape; the faces of a rectangular prism are rectangles

what's convex

a shape in the plane or in space is convex if any line segment connecting two points on the shape lie entirely within the shape,

what's a cylinder

a tube shaped object

how are prisms named

according to the kind of polygons that make up the bases of the prism

given a polyhedron, how can students study its characteristics and qualities?

by noticing its faces, edges, and vertices what kinds of shapes are the faces how many faces are there how many vertices and how many edges does the shape have

where does the name polyhedron come from

comes from the greek; it makes sense because poly means many and hedron means base, so that polyhedron means many bases

just as cylinders are shapes that are related to prisms, what shape is related to pyramids

cones

what are cones

cones are objects like ice cream cones or cone shaped paper cups

describe the different kinds of solid shapes there are...

curved outer surfaces, others have flat surfaces consisting of triangles, squares, rectangles, pentagons, or other polygons

what shape is related to prisms

cylinder

related to prisms and pyramids are...

cylinders and cones

the platonic solids are the only convex polyhedra having what two properties

each one is made of only one kind of regular polygon, and the same number of polygons come together at each vertex

why are platonic solids special

each one is made of only one kind of regular polygon, and the same number of polygons come together at each vertex

describe the relationships between right prisms and all prisms

every right prism is a prism

what's a common error when it comes to naming prisms

ex: a common error is to call a rectangular prism a rectangle

what are platonic solids

five special polyhedra that are very regular and uniform all around

what's a dodecahedron

has 12 regular pentagon faces, with 3 pentagons coming together at each vertex

what's an icosahedron

has 20 equilateral faces, with 5 triangles coming together at each vertex

what's a tetrahedron

has 4 equilateral triangle faces, with 3 triangles coming together at each vertex

what's a cube

has 6 square faces, with 3 squares coming together at each vertex

what's a octahedron

has 8 equilateral triangle faces, with 4 triangles coming together at each vertex

where does the name platonic solids come from

hedron comes from the greek for base or seat, while tetra, octa, dodeca, and icosa means 4, 8, 12, and 20

convex polyhedra don't have any

intrusions or indentations

is a prism is moved to a different orientation in space...

it is still a prism...when a prism is rotated sideways, it is still a prism

what's the base of a pyramid

original polygon

who thought of these solid shapes as associated with earth, fire, water, air, and the whole universe

plato

what's the plural of polyhedron

polyhedra

one special type of polyhedron is a ...

prism

what are the two common types of polyhedra

prisms and pyramids

what's another type of special polyhedron, in addition to prisms

pyramid

what's a right pyramid

pyramid for which the point lies straight up over the center of the base

prism with a rectangle base is

rectangular prism

what are polyhedra

solid shapes with flat surfaces consisting of triangles, squares, rectangles, pentagons, or other polygons

for a polyhedron to be convex..

the angles coming together at a vertex must add to at most 360 degrees, and this puts limits on the convex polyhedra that are possible

what are the edges of the polyhedron

the places where two faces come together

what are the faces of the polyhedron

the polygons that make up the outer surface of the polyhedron

what are the bases of the cylinder

the regions formed by the two starting curves, in some cases, you want to consider the two bases as part of the cylinders

give an example of a cylinder

the tube inside a roll of paper towels

what are the bases of the right prism

the two polygons that you started with

are cylinders and cones polyhedra? if no, why?

they have curved surfaces

ex: a prism with a triangle base is a

triangular prism

if you make the platonic solids with plastic or paper polygons, you may begin to see why there are only five special types of platonic solids...explain

when you put two polygons together to make polyhedrons, you may notice that when the angles at a vertex add to more than 360 degrees, the shape you make will be forced to have indentations/protrusions, and therefore will not be convex

what's the difference between a right cylinder and an oblique cylinder

whether one base is or is not straight up over the other