Chapter 13: Solid Shapes and their Volume & Surface

AreaSection 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 crystalScattergories

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

• 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.

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 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