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11.2 – What Is a Polyhedron?
Key Terms
 Bases – Faces of a geometric solid from which the height is drawn.
 Cube – A rectangular prism whose six faces are congruent squares.
 The edges of a cube all have the same length.
 Edges – The line segments bordering each face of a solid geometric figure.
 Faces – The plane figures that make up the surfaces of a solid geometric figure.
 Lateral Faces – The plane figures that make up the surfaces of a solid geometric figure that are not bases of the figure.
 Polyhedron – A solid figure with no curved surfaces or edges.
 The faces are polygons, the edges are line segments, and the points at which the edges meet are called the vertices.
 Prism – A threedimensional solid consisting of two parallel congruent polygons and all the points between them.
 Pyramid – A threedimensional solid consisting of a base that can be any polygon, a point not in the same plane as the polygon, and all the points between them.
Review
Polygons 
 They are twodimensional.
 They are closed (no gaps between sides).
 All of their sides are segments (no curves).

Notes
Prisms 
 A prism is a polyhedron (solid) made of two parallel congruent polygons and all the points between them.
 Bases: the two parallel congruent faces (like a top and bottom)
 Lateral Faces: all rectangles that are not bases (like sides)
 Altitude: the line segment whose length is the height of a prism

 Compare a Line Segment and Prism
 Line segment: two points and all the points between them
 Prism: two bases and all the points between them
 Ex 1. For the shape of prisms, why can any two opposite faces of a rectangular prism be called the bases?
 Answer: All the faces are rectangles.
 Ex 2. Why is a cube a special kind of rectangular prism?
 Answer: All its faces are congruent rectangles.


Pyramids 
 A polyhedron (solid) made of one polygon base, a point not on the same plane, and all the points between them.
 Bases: one polygon face (like a bottom)
 Lateral Faces: all triangles that meet at the same point (like sides)
 Altitude: the line segment whose length is the bases of a pyramid
 Pyramid Bases and Faces
 Triangular pyramids have triangular bases, so any face on a triangular pyramid can be treated as the base.
 Some pyramids have bases made of other polygons (see below).
 Regular & Irregular Pyramids
 Irregular pyramids have irregular polygon bases

Item 
Regular Pyramid 
Irregular Pyramid 
Base 
Regular Polygon 
Irregular Polygon 
Lateral Faces 
Congruent Triangles 
Not Congruent Triangles 
Vertex 
Centered Over Base 
Not Centered Over Base 
Altitude 
Through Center 
Not Through Center 

 Compare a Line Segment to a Pyramid
 Line segment: two points and all the points between them
 Pyramid: a polygon and a point not on the same plane and all the points between themWhy is a triangular pyramid a special kind of pyramid?
 Answer: Any face can be treated as the base, since all the faces are triangles.
 How are oblique pyramids and right pyramids different?
 Answer: Oblique pyramids are slanted; in a right pyramid, the vertex is centered over the base.Building Polyhed gif

Piling 
 Another way to form a prism is to pile congruent shapes on top of each other like a deck of cards.
 For prisms, each shape the same size and shape (as you pile them).
 For pyramids, each shape is the same shape, but a smaller size (as you pile them).
 Pyramids have a vertex (point) at the top (no size or shape); and therefore, the vertex cannot be similar to the base.
 Congruent vs Similar Bases
 All crosssectional shapes that are parallel or perpendicular to one of the bases of a prism are also congruent to one another.
 A pyramid has crosssectional shapes, taken parallel to its base, that are similar to one another.

Examples
 Ex 1. Which of the following terms correctly describe the object below?
Answers: solid, cube, prism

 Ex 2. Which of the following terms correctly describe the object below?
Answers: pyramid, polyhedron, solid

 Ex 3. It is true that the vertical crosssectional shapes of this prism are all congruent triangles.

Important!
Practice (Apex Study 11.2)
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