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Chapter 13: Solid Shapes and their Volume & Surface AreaPowerPoint Presentation

Chapter 13: Solid Shapes and their Volume & Surface Area

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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
- 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.

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

- Find the number of and describe the faces of the following icosidodecahedron, and then find the number of edges and vertices.

Non-Examples

- Spheres and cylinders are not polyhedral because their surfaces are not made of polygons.

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

- 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

- Adding a pyramid to each pentagon of an icosidodecahedron creates a new polyhedron with 80 triangular faces called a pentakisicosidodecahedron.

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

- A cone consists of a closed curve, a point in a different plane, and a surface joining the point to the curve

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

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

- http://folk.uib.no/nmioa/kalender/

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

- 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

- Def: The height of a prism or cylinder is the perpendicular distance between the planes containing the bases

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

- 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

- Def: The height of a pyramid or cone is the perpendicular length between the apex and the base.

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

- Ex 3: Calculate the volume of the
following octahedron.

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

- 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

- 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

- Archimedes’s Principle: An object that floats displaces the amount of water that weighs as much as the object

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