# 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

• 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!
• Cylinders and Prisms
• The base of a cylinder is a circle (disc)
• Cylinders are NOT polyhedrons
• The base of a prism is a polygon

• 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.

• 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

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!
• Cones and Pyramids
• The base of a cone is a circle (disc)
• Cones are NOT polyhedrons
• The base of a pyramid is a polygon
• 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).

• 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.

• 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.

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

## 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)