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**Chapter 13: Solid Shapes and their Volume & Surface Area**

Section 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 crystal Scattergories 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**

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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 𝑉𝑜𝑙𝑢𝑚𝑒= 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 ∙ℎ𝑒𝑖𝑔ℎ𝑡 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 𝑙, width 𝑤, and height ℎ is 𝑉𝑜𝑙𝑢𝑚𝑒=𝑙∙𝑤∙ℎ Ex 2: The volume of a circular cylinder with the radius of the base being 𝑟 and height ℎ is 𝑉𝑜𝑙𝑢𝑚𝑒=𝜋∙ 𝑟 2 ∙ℎ

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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 𝑉𝑜𝑙𝑢𝑚𝑒= 1 3 ∙(𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒)∙ℎ𝑒𝑖𝑔ℎ𝑡 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 𝑟 is given by 𝑉𝑜𝑙𝑢𝑚𝑒= 4 3 ∙𝜋∙ 𝑟 3 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 𝑐 𝑚 3 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**

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