Presentation on theme: "Chapter 13: Solid Shapes and their Volume & Surface Area Section 13.1: Polyhedra and other Solid Shapes."— Presentation transcript:
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 PyramidIcosidodecahedron
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 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
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 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
Volumes of Particular Prisms and Cylinders
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
Volume Example Ex 3: Calculate the volume of the following octahedron.
Volume of a Sphere
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
Volume of Objects that Float Archimedes’s Principle: An object that floats displaces the amount of water that weighs as much as the object