Download 8.4

Chapter 8: Informal Geometry
Section 8.4: Simple Closed Surfaces
Visualizing Three-Dimensional Figures

Simple closed surface: a defined region in space without any holes through it; must separate the points
of space into three disjoint sets of points: interior, exterior, and surface

Solid: union of the points interior to a simple closed surface and the surface

Space region: solids are sometimes called space regions

Which of these are simple closed surface solids?
A.
B.
C.
D.
E.
Answer: B and E
Polyhedra

Polyhedron: a polyhedron is a simple closed surface in space whose boundary is composed of
polygonal regions
Which of these are NOT polyhedrons?

Answer: Cone, cylinder, and sphere
Prism: Polygons with the same size and shape are congruent; A prism is a polyhedron formed by two
congruent polygonal regions in parallel planes, along with three or more regions bounded by
parallelograms joining the two polygonal regions so as to form a closed surface.

Bases: the parallel polygonal regions of a prism

Lateral edges: parallel edges joining the two bases

Lateral faces: regions bounded by parallelograms
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Faces: bases and lateral faces are ALL called faces

Right prism: the lateral faces are rectangular regions and the planes containing the lateral faces are
perpendicular to the planes containing the bases

Oblique prism: the lateral faces are not all rectangular regions and the planes containing the lateral
faces are not perpendicular to the planes containing the bases

Naming a prism: prisms get their names from their bases

Triangular prism: base is a triangle; lateral faces are rectangles

Quadrilateral prism: base is a quadrilateral; lateral faces are rectangles
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Hexagonal prism: base is a hexagon; lateral faces are rectangles

Pyramid: a polyhedron formed by a simple closed polygonal region (called the base) a point (called the
vertex or the apex) not in the plane of the region, and the triangular regions joining the point and the
sides of the polygonal region.

Regular pyramid: the base is a regular polygon; all of the lateral faces are all congruent triangles

Naming a pyramid: pyramids get their names from their bases
Name the regular pyramids shown above: triangular pyramid, decagonal pyramid, hexagonal pyramid,
pentagonal pyramid, hexagonal pyramid, square pyramid
Euler’s Formula

Vertices: use the letter V to represent vertices in the formula

Faces: use the letter F to represent faces in the formula

Edges: use the letter E to represent edges in the formula

Euler’s formula: V + F – E = 2; 2 is sometimes referred to as “Euler’s Constant”
Regular Polyhedra [Platonic Solids]
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Regular polyhedron: if the faces of a convex polyhedron are congruent regular polygonal regions, and
if each vertex is the intersection of the same number of edges, then the polyhedron is called a regular
polyhedron. There are exactly five of these.

Tetrahedron: formed by four congruent triangular regions

Cube: formed by six congruent square regions; the cube is also known as a hexahedron

Octahedron: formed by eight congruent triangular regions

Dodecahedron: formed by twelve congruent pentagonal regions

Icosahedron: formed by twenty congruent triangular regions
Yup! You guessed it!  D&D dice are in the shapes of the five regular polyhedrons! 
Dihedral Angles

Dihedral angle: if two planes intersect, a dihedral angle is the union of the two noncoplanar half planes
and the line of intersection.

Faces: the two half planes of the dihedral angle

Edge: the common line of the dihedral angle

Plane angle of the dihedral angle: a plane angle formed by two rays, one in each face of a dihedral
angle, with each ray having its endpoint on the edge of the dihedral angle and each ray perpendicular to
the edge
Spheres, Cylinders, and Cones

Sphere: a set of all points in space at a given distance from a fixed point is called a sphere

Center: the fixed point of the sphere

Radius: any line segment from the center to a point on the sphere

Interior: the set of all points whose distance from the center of the sphere is less than the measure of the
radius

Exterior: the set of all points whose distance from the center of the sphere is greater than the measure of
the radius

Tangent: if the intersection between a plane and a sphere is a single point, the plane is tangent to the
sphere

Circular cylinder: formed by two parallel planes intersecting a sphere and the line segments connecting
the circular regions by their edges such that every perpendicular planar cross section of the cylinder
would be a circular region; the bases of the cylinder are circles, while the lateral face of the cylinder is a
quadrilateral

Right circular cylinder: analogous to a prism, the bases are at right angles to the lateral face, making
the lateral face a rectangle; the dimensions of this rectangle are 2𝜋r by r, where r is the radius.

Circular cone: consists of a circular base and segments joining a fixed point (vertex) to points on the
base.
Cone
Cone with half sphere – yummy! 
Exercise Sets:
Homework:
p. 415: 1abc, 2ad, 3aceg, 4, 5, 6, 9, 11aceg, 16ac, 17ace, 18, 21, 23
Geogebra:
p. 415: 7ac, 10ac
Blazeview:
p. 415: 8, 12ac, 13bd, 19