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.

Examples of Polyhedra Cube Pyramid Icosidodecahedron

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)

Prism Examples

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

Pyramid Examples

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.

See Activity 13B

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

Platonic Solids Pyrite crystal Scattergories die

Section 13.2: Patterns and Surface Area

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

Section 13.3: Volumes of Solid Shapes

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 𝑉𝑜𝑙𝑢𝑚𝑒= 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 ∙ℎ𝑒𝑖𝑔ℎ𝑡 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 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

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

See examples problem in Activity 13N

Section 13.4: Volumes 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 𝑐 𝑚 3 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

See example problems in Activity 13Q