1. Chapter 13: Solid Shapes and their Volume & Surface Area
Section 13.1: Polyhedra and other Solid Shapes

2. 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.

3. Examples of Polyhedra
Cube Pyramid Icosidodecahedron

4. Example 1
Find the number of and describe the faces of the following octahedron, and then find the number of edges and vertices.

5. Example 2
Find the number of and describe the faces of the following icosidodecahedron, and then find the number of edges and vertices.

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

7. 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)

8. Prism Examples

9. 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

10. Pyramid Examples

11. 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.

12. See Activity 13B

13. 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

14. 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

15. 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

16. Platonic Solids
Pyrite crystal
Scattergories
die

17. Section 13.2: Patterns and Surface Area

18. 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)

19. How to create a dodecahedron calendar

20. 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:

21. Section 13.3: Volumes of Solid Shapes

22. 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

23. Volumes of Prisms and Cylinders
Def: The height of a prism or cylinder is the perpendicular distance between the planes containing the bases

24. 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.

25. 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

26. Volumes of Pyramids and Cones
Def: The height of a pyramid or cone is the perpendicular length between the apex and the base.

27. 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

28. Volume Example
Ex 3: Calculate the volume of the
following octahedron.

29. 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.

30. 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.

31. See examples problem in Activity 13N

32. Section 13.4: Volumes of Submerged Objects

33. 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?

34. Volume of Objects that Float
Archimedes’s Principle: An object that floats displaces the amount of water that weighs as much as the object

35. See example problems in Activity 13Q