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)
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
A very complicated example • Adding a pyramid to each pentagon of an icosidodecahedron creates a new polyhedron with 80 triangular faces called a pentakisicosidodecahedron.
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
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
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.
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