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Chapter 13: Solid Shapes and their Volume & Surface Area. Section 13.1: Polyhedra and other Solid Shapes. Basic Definitions. A polyhedron is a closed, connected shape in space whose outer surfaces consist of polygons

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chapter 13 solid shapes and their volume surface area

Chapter 13: Solid Shapes and their Volume & Surface Area

Section 13.1: Polyhedra and other Solid Shapes

basic definitions
Basic Definitions
  • A polyhedron is a closed, connected shape in space whose outer surfaces consist of polygons
  • A face of a polyhedron is one of the polygons that makes up the outer surface
  • An edge is a line segment where two faces meet
  • A vertex is a corner point where multiple faces join together
  • Polyhedra are categorized by the numbers of faces, edges, and vertices, along with the types of polygons that are faces.
examples of polyhedra
Examples of Polyhedra

Cube

Pyramid Icosidodecahedron

example 1
Example 1
  • Find the number of and describe the faces of the following octahedron, and then find the number of edges and vertices.
example 2
Example 2
  • Find the number of and describe the faces of the following icosidodecahedron, and then find the number of edges and vertices.
non examples
Non-Examples
  • Spheres and cylinders are not polyhedral because their surfaces are not made of polygons.
special types of polyhedra
Special Types of Polyhedra
  • A prism consists of two copies of a polygon lying in parallel planes with faces connecting the corresponding edges of the polygons
  • Bases: the two original polygons
  • Right prism: the top base lies directly above the

bottom base without any twisting

  • Oblique prism: top face is shifted instead of

being directly above the bottom

  • Named according to its base (rectangular prism)
more special polyhedra
More Special Polyhedra
  • A pyramid consists of a base that is a polygon,

a point called the apex that lies on a different

plane, and triangles that connect the apex to

the base’s edges

  • Right pyramid: apex lies directly above the

center of the base

  • Oblique pyramid: apex is not above the center
a very complicated example
A very complicated example
  • Adding a pyramid to each pentagon of an icosidodecahedron creates a new polyhedron with 80 triangular faces called a pentakisicosidodecahedron.
similar solid shapes
Similar Solid Shapes
  • A cylinder consists of 2 copies of a closed curve (circle, oval, etc) lying in parallel planes with a 2-dimensional surface wrapped around to connect the 2 curves
  • Right and oblique cylinders are defined similarly to those of prisms
other similar solid shapes
Other Similar Solid Shapes
  • A cone consists of a closed curve, a point in a different plane, and a surface joining the point to the curve
platonic solids
Platonic Solids
  • A Platonic Solid is a polyhedron with each face being a regular polygon of the same number of sides, and the same number of faces meet at every vertex.
  • Only 5 such solids:
    • Tetrahedron: 4 equilateral triangles as faces, 3 triangles meet at each vertex
    • Cube: 6 square faces, 3 meet at each vertex
    • Octahedron: 8 equilateral triangles as faces, 4 meet at each vertex
    • Dodecahedron: 12 regular pentagons as faces, 3 at each vertex
    • Icosahedron: 20 equilateral triangles as faces, 5 at each vertex
platonic solids1
Platonic Solids

Pyrite crystal

Scattergories

die

making polyhedra from 2 dimensional surfaces
Making Polyhedra from 2-dimensional surfaces
  • Many polyhedral can be constructed by folding and joining two-dimensional patterns (called nets) of polygons.
  • Helpful for calculating surface area of a 3-D shape, i.e. the total area of its faces, because you can add the areas of each polygon in the pattern (as seen on the homework)
how to create a dodecahedron calendar
How to create a dodecahedron calendar
  • http://folk.uib.no/nmioa/kalender/
cross sections
Cross Sections
  • Given a solid shape, a cross-section of that shape is formed by slicing it with a plane.
  • The cross-sections of polyhedral are polygons.
  • The direction and location of the plane can result in several different cross-sections
  • Examples of cross-sections of the cube: https://www.youtube.com/watch?v=Rc8X1_1901Q
definitions and principles
Definitions and Principles
  • Def: The volume of a solid shape is the number of unit cubes that it takes to fill the shape without gap or overlap
  • Volume Principles:
    • Moving Principle: If a solid shape is moved rigidly without stretching or shrinking it, the volume stays the same
    • Additive Principle: If a finite number of solid shapes are combined without overlap, then the total volume is the sum of volumes of the individual shapes
    • Cavalieri’s Principle: The volume of a shape and a shape made by shearing (shifting horizontal slices) the original shape are the same
volumes of prisms and cylinders
Volumes of Prisms and Cylinders
  • Def: The height of a prism or cylinder is the perpendicular distance between the planes containing the bases
volumes of prisms and cylinders1
Volumes of Prisms and Cylinders
  • Formula: For a prism or cylinder, the volume is given by
  • The formula doesn’t depend on whether the shape is right or oblique.
volumes of particular prisms and cylinders
Volumes of Particular Prisms and Cylinders
  • Ex 1: The volume of a rectangular box with length , width , and height is
  • Ex 2: The volume of a circular cylinder with the radius of the base being and height is
volumes of pyramids and cones
Volumes of Pyramids and Cones
  • Def: The height of a pyramid or cone is the perpendicular length between the apex and the base.
volumes of pyramids and cones1
Volumes of Pyramids and Cones
  • Formula:For a pyramid or cone, the volume is given by
  • Again, the formula works whether the shape is right or oblique
volume example
Volume Example
  • Ex 3: Calculate the volume of the

following octahedron.

volume of a sphere
Volume of a Sphere
  • Formula: The volume of a sphere with radius is given by
  • See Activity 13O for explanation of why this works.
volume vs surface area
Volume vs. Surface Area
  • As with area and perimeter, increasing surface area generally increases volume, but not always.
  • With a fixed surface area, the cube has the largest volume of any rectangular prism (not of any polyhedron) and the sphere has the largest volume of any 3-dimensional object.
volume of submerged objects
Volume of Submerged Objects
  • The volume of an 3-dimensional object can be calculated by determining the amount of displaced liquid when the object is submerged.
  • Ex: If a container has 500 mL of water in it, and the water level rises to 600 mL after a toy is submerged, how many is the volume of the toy?
volume of objects that float
Volume of Objects that Float
  • Archimedes’s Principle: An object that floats displaces the amount of water that weighs as much as the object