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Chapter 13: Solid Shapes and their Volume & Surface
AreaSection 13.1: Polyhedra and other Solid Shapes
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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.
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Examples of Polyhedra Cube
Pyramid Icosidodecahedron
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Example 1
• Find the number of and describe the faces of the following octahedron, and then find the number of edges and vertices.
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Example 2
• Find the number of and describe the faces of the following icosidodecahedron, and then find the number of edges and vertices.
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Non-Examples
• Spheres and cylinders are not polyhedral because their surfaces are not made of polygons.
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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)
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Prism Examples
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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
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Pyramid Examples
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A very complicated example
• Adding a pyramid to each pentagon of an icosidodecahedron creates a new polyhedron with 80 triangular faces called a pentakis icosidodecahedron.
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See Activity 13B
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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
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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
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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
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Platonic Solids
Pyrite crystalScattergories
die
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Section 13.2: Patterns and Surface Area
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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)
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How to create a dodecahedron calendar• http://folk.uib.no/nmioa/kalender/
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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
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Section 13.3: Volumes of Solid
Shapes
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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
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Volumes of Prisms and Cylinders
• Def: The height of a prism or cylinder is the perpendicular distance between the planes containing the bases
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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.
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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
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Volumes of Pyramids and Cones
• Def: The height of a pyramid or cone is the perpendicular length between the apex and the base.
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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
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Volume Example
• Ex 3: Calculate the volume of the following octahedron.
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Volume of a Sphere
• Formula: The volume of a sphere with radius is given by
• See Activity 13O for explanation of why this works.
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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.
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See examples problem in Activity 13N
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Section 13.4: Volumes of
Submerged Objects
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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?
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Volume of Objects that Float
• Archimedes’s Principle: An object that floats displaces the amount of water that weighs as much as the object
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See example problems in Activity 13Q