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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 three-dimensional solid consisting of two parallel congruent polygons and all the points between them.
  • Pyramid – A three-dimensional 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 two-dimensional.
  • They are closed (no gaps between sides).
  • All of their sides are segments (no curves).

Notes

Polyhedrons
  • Three-dimensional solid
  • Bounded by polygons regions formed by intersecting planes.
    • These planes are called faces
  • Encloses a single region of space
    • All one- and two-dimensional objects exist in a plane. And all planes exist in space.
    • In other words, all objects in geometry lie in space.
  • A polyhedron is contained by four or more intersecting planes.
  • Polyhedrons can be pyramids, prisms, or other combinations of polygons.
    • Polyhedrons are never both, prisms and pyramids.

GeoB 11.2 Polyhedrons


GeoB 11.2 Polyhedron Parts

  • Every polyhedron is made of the following elements
    • Faces, which are the plane figures that make its surfaces.
    • Edges , which are the line segments bordering each face.
    • Vertices, which are the points at the ends of each edge.
GeoB 11.2 PolyhedronFaces GeoB 11.2 PolyhedronEdges GeoB 11.2 PolyhedronVertices
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

GeoB 11.2 Prism Oblique Right

  • 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.
  • Venn Diagram

GeoB 11.2 VennD

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

GeoB 11.2 Pyramids


  • 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).

GeoB 11.2 Pyramids Faces


  • Regular & Irregular Pyramids
    • Irregular pyramids have irregular polygon bases

GeoB 11.2 Reg Irreg Pyramid

  • Chart
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 cross-sectional shapes that are parallel or perpendicular to one of the bases of a prism are also congruent to one another.
    • A pyramid has cross-sectional shapes, taken parallel to its base, that are similar to one another.

GeoB 11.2 Piling

GeoB 11.2 Building Polyhedrons from Piles

Examples

  • Ex 1. Which of the following terms correctly describe the object below?

GeoB 11.2 Ex1

Answers: solid, cube, prism

  • Ex 2. Which of the following terms correctly describe the object below?

GeoB 11.2 Ex2

Answers: pyramid, polyhedron, solid

  • Ex 3. It is true that the vertical cross-sectional shapes of this prism are all congruent triangles.

GeoB 11.2 Ex3


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