Presentation on theme: "Chapter 13: Solid Shapes and their Volume & Surface Area"— Presentation transcript:
1 Chapter 13: Solid Shapes and their Volume & Surface Area Section 13.1: Polyhedra and other Solid Shapes
2 Basic DefinitionsA polyhedron is a closed, connected shape in space whose outer surfaces consist of polygonsA face of a polyhedron is one of the polygons that makes up the outer surfaceAn edge is a line segment where two faces meetA vertex is a corner point where multiple faces join togetherPolyhedra are categorized by the numbers of faces, edges, and vertices, along with the types of polygons that are faces.
3 Examples of PolyhedraCube Pyramid Icosidodecahedron
4 Example 1Find the number of and describe the faces of the following octahedron, and then find the number of edges and vertices.
5 Example 2Find the number of and describe the faces of the following icosidodecahedron, and then find the number of edges and vertices.
6 Non-ExamplesSpheres and cylinders are not polyhedral because their surfaces are not made of polygons.
7 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 polygonsBases: the two original polygonsRight prism: the top base lies directly above thebottom base without any twistingOblique prism: top face is shifted instead ofbeing directly above the bottomNamed according to its base (rectangular prism)
9 More Special Polyhedra A pyramid consists of a base that is a polygon,a point called the apex that lies on a differentplane, and triangles that connect the apex tothe base’s edgesRight pyramid: apex lies directly above thecenter of the baseOblique pyramid: apex is not above the center
13 Similar Solid ShapesA 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 curvesRight and oblique cylinders are defined similarly to those of prisms
14 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
15 Platonic SolidsA 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 vertexCube: 6 square faces, 3 meet at each vertexOctahedron: 8 equilateral triangles as faces, 4 meet at each vertexDodecahedron: 12 regular pentagons as faces, 3 at each vertexIcosahedron: 20 equilateral triangles as faces, 5 at each vertex
18 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)
20 Cross SectionsGiven 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-sectionsExamples of cross-sections of the cube: https://www.youtube.com/watch?v=Rc8X1_1901Q
22 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 overlapVolume Principles:Moving Principle: If a solid shape is moved rigidly without stretching or shrinking it, the volume stays the sameAdditive Principle: If a finite number of solid shapes are combined without overlap, then the total volume is the sum of volumes of the individual shapesCavalieri’s Principle: The volume of a shape and a shape made by shearing (shifting horizontal slices) the original shape are the same
23 Volumes of Prisms and Cylinders Def: The height of a prism or cylinder is the perpendicular distance between the planes containing the bases
24 Volumes of Prisms and Cylinders Formula: For a prism or cylinder, the volume is given by𝑉𝑜𝑙𝑢𝑚𝑒= 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 ∙ℎ𝑒𝑖𝑔ℎ𝑡The formula doesn’t depend on whether the shape is right or oblique.
25 Volumes of Particular Prisms and Cylinders Ex 1: The volume of a rectangular box with length 𝑙, width 𝑤, and height ℎ is𝑉𝑜𝑙𝑢𝑚𝑒=𝑙∙𝑤∙ℎEx 2: The volume of a circular cylinder with the radius of the base being 𝑟 and height ℎ is𝑉𝑜𝑙𝑢𝑚𝑒=𝜋∙ 𝑟 2 ∙ℎ
26 Volumes of Pyramids and Cones Def: The height of a pyramid or cone is the perpendicular length between the apex and the base.
27 Volumes of Pyramids and Cones Formula: For a pyramid or cone, the volume is given by𝑉𝑜𝑙𝑢𝑚𝑒= 1 3 ∙(𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒)∙ℎ𝑒𝑖𝑔ℎ𝑡Again, the formula works whether the shape is right or oblique
28 Volume ExampleEx 3: Calculate the volume of thefollowing octahedron.
29 Volume of a SphereFormula: The volume of a sphere with radius 𝑟 is given by𝑉𝑜𝑙𝑢𝑚𝑒= 4 3 ∙𝜋∙ 𝑟 3See Activity 13O for explanation of why this works.
30 Volume vs. Surface AreaAs 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.
33 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?
34 Volume of Objects that Float Archimedes’s Principle: An object that floats displaces the amount of water that weighs as much as the object