11 4 Geometry in Three Dimensions Simple closed

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11 -4 Geometry in Three Dimensions • Simple closed surfaces and polyhedra. • How

11 -4 Geometry in Three Dimensions • Simple closed surfaces and polyhedra. • How to draw three-dimensional shapes. • Why there are only five regular polyhedra. • Simple closed surfaces that are not polyhedra. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 1

Simple Closed Surfaces A simple closed surface has exactly one interior, no holes, and

Simple Closed Surfaces A simple closed surface has exactly one interior, no holes, and is hollow. These two figures are not simple, closed surfaces. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 2

Simple Closed Surfaces Sphere – the set of all points at a given distance

Simple Closed Surfaces Sphere – the set of all points at a given distance from a given point (the center) Solid – the set of all points on a simple closed surface together with all interior points Polyhedron – a simple, closed surface made up of polygonal regions (faces). The vertices of the polygonal regions are the vertices of the polyhedron. The sides of the polygonal regions are called the edges of the polyhedron. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 3

Polyhedra: Not polyhedra: ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 4

Polyhedra: Not polyhedra: ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 4

Prisms A prism is a polyhedron in which two congruent faces (the bases) lie

Prisms A prism is a polyhedron in which two congruent faces (the bases) lie in parallel planes and the other faces (lateral faces) are bounded by parallelograms. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 5

Prisms Right prism – lateral faces are perpendicular to the bases. Oblique prism –

Prisms Right prism – lateral faces are perpendicular to the bases. Oblique prism – lateral faces are not perpendicular to the bases. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 6

Pyramids A pyramid is a polyhedron determined by a polygon and a point not

Pyramids A pyramid is a polyhedron determined by a polygon and a point not in the plane of the polygon. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 7

Pyramids Right pyramid – lateral faces are congruent isosceles triangles. ALWAYS LEARNING Copyright ©

Pyramids Right pyramid – lateral faces are congruent isosceles triangles. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 8

Truncated Polyhedron If one or more corners of a polyhedron is removed by an

Truncated Polyhedron If one or more corners of a polyhedron is removed by an intersecting plane or planes, the polyhedron is a truncated polyhedron. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 9

Regular Polyhedra Convex polyhedron – a polyhedron in which a segment connecting any two

Regular Polyhedra Convex polyhedron – a polyhedron in which a segment connecting any two points in the interior of the polyhedron is in the interior of the polyhedron. Regular polyhedron – a convex polyhedron whose faces are congruent regular polygonal regions such that the number of edges that meet at each vertex is the same for all the vertices of the polyhedron. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 10

Regular Polyhedra Cube ALWAYS LEARNING Tetrahedron Octahedron Copyright © 2020, 2016, 2012 Pearson Education,

Regular Polyhedra Cube ALWAYS LEARNING Tetrahedron Octahedron Copyright © 2020, 2016, 2012 Pearson Education, Inc. 11

Regular Polyhedra Dodecahedron ALWAYS LEARNING Icosahedron Copyright © 2020, 2016, 2012 Pearson Education, Inc.

Regular Polyhedra Dodecahedron ALWAYS LEARNING Icosahedron Copyright © 2020, 2016, 2012 Pearson Education, Inc. 12

ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 13

ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 13

ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 14

ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 14

Nets These patterns can be used to construct the five regular polyhedra. ALWAYS LEARNING

Nets These patterns can be used to construct the five regular polyhedra. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 15

Cylinders and Cones Cylinder – a simple, closed surface that is not a polyhedron;

Cylinders and Cones Cylinder – a simple, closed surface that is not a polyhedron; formed as a segment AB parallel to a given line traces a planar curve other than a polygon. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 16

Cylinders and Cones Cone – the union of the line segments connecting a point

Cylinders and Cones Cone – the union of the line segments connecting a point P with each point of a simple, closed curve, the simple, closed curve, and the interior of the curve. ALWAYS LEARNING Copyright © 2020, 2016, 2012 Pearson Education, Inc. 17

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