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 Scattergories die Pyrite crystal

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 𝑐𝑚 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