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11.3 – Cylinders and Cones

Key Terms

  • Cylinder – A three-dimensional solid made from two congruent and parallel circular bases and all the points between them.
  • Cone – A three-dimensional solid made from one circular base, a vertex (not in the same plane), and all the points (space) between them.

Review

Polyhedrons
  • Prisms and pyramids are polyhedrons because all of their faces are polygons

GeoB 11.3 Cone Prism Cylinder Prism Base

  • Altitudes (the heights of solids) are always perpendicular (90 degrees) to the bottom base

Notes

Cylinders
  • Properties of a Cylinder
    • It is a solid
    • It has two circular bases (discs)
    • The two bases congruent
    • The two bases parallel
    • The bases lie in different planes
    • Can be right or oblique
      • In an oblique prism, the bases are not centered over each other.  It’s the same for oblique cylinders!
GeoB 11.3 Cylinder Oblique
  • Cylinders and Prisms
    • The base of a cylinder is a circle (disc)
      • Cylinders are NOT polyhedrons
    • The base of a prism is a polygon
 GeoB 11.3 Cylinder Prism
  • Construction of a Cylinder
    • A cylinder has one lateral face (a rectangle) and two congruent circular bases.
      • It is like a rectangle that is rolled into a tube between the two circular bases (the top and bottom of the tube).
    • You can also create a cylinder by rotating a rectangle along its line of symmetry.
      • Imagine this rectangle (below) spinning very fast on the y-axis.  You would see the shape of a cylinder form.

GeoB 11.3 Animate Rectangle Spin

  • Piling of Circles to Make Cylinders
    • A cylinder, like a prism, can also be thought of as a pile of congruent two-dimensional shapes (in this case, circles / discs).
    • These circles are horizontal cross-sections of the cylinder.
    • Example: a stack of CDs

GeoB 11.3 Cylinder Pile

 

Cones
  • Properties of a Cone
    • It is a solid
    • It has one circular base (disc)
    • It has a vertex located in a different plane than the base
    • Can be right or oblique
      • In an oblique pyramid, the vertex is not centered over its base.  It’s the same for oblique cones!
GeoB 11.3 Cone Parts
  • Cones and Pyramids
    • The base of a cone is a circle (disc)
      • Cones are NOT polyhedrons
    • The base of a pyramid is a polygon
GeoB 11.3 Cone Prism
  • Construction of a Cone
    • The lateral face of a cone is like a section of a circle curved around the circular base.
      • These are the “sectors of a circle” you learned how to measure the area of during the “Circle” unit.
    • The smaller your section of circle is for the lateral face of a cone, the pointier your cone will be.
      • The 90° sector makes a tall, pointy cone (like a hat or ice cream cone).
      • The 180° sector makes a wider cone (like a waffle cone).
      • The 170° sector makes a very wide cone (like the roof of a cylindrical tower).

GeoB 11.3 Cone Construction

  • You can also create a cone by rotating a triangle along its line of symmetry.
    • Imagine this triangle (below) spinning very fast on the y-axis. You would see the shape of a cone form.

GeoB 11.3 Rotate Triangle

  • Piling of Circles to Make Cones
    • A cone, like a pyramid, can also be thought of as a pile of similar two-dimensional shapes (in this case, circles / discs) and a vertex (on top).
    • These circles are horizontal cross-sections of the cone.

GeoB 11.3 Cone Piles

 

Summary
  • Cones and cylinders are NOT polyhedrons because their bases are NOT polygons
  • Pyramids and prisms ARE polyhedrons because their bases ARE polygons
  • Cones and pyramids have SIMILAR parallel horizontal cross-sections and only ONE base
  • Cylinders and prisms have CONGRUENT parallel horizontal cross-sections and TWO bases
  • Cones and pyramids have vertices (which are NOT considered a one of the similar horizontal cross-sections of the solid

GeoB 11.3 Compare 4

GeoB 11.3 Cone Prism Cylinder Prism

Examples

  • Ex 1. Using the piling method, it is true that the following can be constructed from polygons alone:
    • Prisms
    • Pyramids (not including the vertex)
  • Ex 2. Using the piling method, it is true that the following can be constructed from discs alone:
    • Cylinders
    • Cones (not including the vertex)
  • Ex 3. Using the piling method, it is true that the following can be constructed from polygons alone or discs alone:
    • Cubes
    • Prisms
    • Cylinders
    • Pyramids (not including the vertex)
    • Cones (not including the vertex)

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