Chapter 11 Part 2. Measurement of Figures and Solids
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1 Chapter 11 Part 2 Measurement of Figures and Solids
2 11.5 Explore Solids Objective: Identify Solids Essential Question: When is a solid a polyhedron?
3 Using properties of polyhedra A is a solid that is bounded by polygons called, that enclose a since region of space. An of a polyhedron is a line segment formed by the intersection of two faces.
4 Using properties of polyhedra A of a polyhedron is a point where three or more edges meet. The plural of polyhedron is or.
5 Polyhedron: a three-dimensional solid made up of plane faces. Prism: a polyhedron (geometric solid) with two parallel, same-size bases joined by 3 or more parallelogramshaped sides. Tetrahedron:
6 Example: Decide whether the solid is a polyhedron. If so, count the number of faces, vertices, and edges of the polyhedron.
7 Types of Solids
8 Regular/Convex/Concave A polyhedron is regular if all its faces are A polyhedron is if any two points on its surface can be connected by a segment that lies entirely or the polyhedron.
9 continued... If this segment goes outside the polyhedron, then the polyhedron is said to be
10 Example: Classifying Polyhedra Is the octahedron convex? Is it regular?
11 Example: Is the octahedron convex? Is it regular?
12 Example: Is the octahedron convex? Is it regular?
13 Imagine a plane slicing through a solid. The intersection of the plane and the solid is called a cross section.
14 Example: Describe the shape formed by the intersection of the plane and the cube.
15 Example: Describe the shape formed by the intersection of the plane and the cube.
16 Using Euler s Theorem:
17 Using Euler s Theorem A cube (6 faces) dodecahedron A regular octahedron (8 faces), icosahedron
18 Notice that the sum of the number of faces and vertices is two more than the number of edges in the solids above. This result was proved by the Swiss mathematician Leonhard Euler. Leonard Euler
19 Euler s Theorem The number of faces (F), vertices (V), and edges (E) of a polyhedron are related by the formula
20 Example: The solid has 14 faces; 8 triangles and 6 octagons. How many vertices does the solid have?
21 Example: In molecules of sodium chloride commonly known as table salt, chloride atoms are arranged like the vertices of regular octahedrons. In the crystal structure, the molecules share edges. How many sodium chloride molecules share the edges of one sodium chloride molecule?
22 Example: SPORTS. A soccer ball resembles a polyhedron with 32 faces; 20 are regular hexagons and 12 are regular pentagons. How many vertices does this polyhedron have?
23 Homework: Exercises 11.5 Regular: #1 32, Honors: #1-32, 34-41
24 11.6 Volume of Prisms and Cylinders Objective: Find volumes of prisms and cylinders. Essential Question: How do you find the volume of a right prism or right cylinder?
25 Exploring Volume The volume of a solid is the number of cubic units contained in its interior. Volume is measured in cubic units, such as cubic meters (m 3 ).
26 Volume Postulates Volume of a cube: Volume Congruence Postulate Volume Addition Postulate:
27 Example: The box shown is 5 units long, 3 units wide, and 4 units high. How many unit cubes will fit in the box? What is the volume of the box?
28 Volume Theorems Volume of a Prism The volume V of a prism is, where B is the area of the base and h is the height. Volume of a Cylinder The volume V of a cylinder is, where B is the area of a base, h is the height, and r is the radius of the base.
29 Example: Find the volume of the right prism.
30 Example: Find the volume of the right cylinder.
31 Example: Use the measurements given to solve for x.
32 Example: Concrete weighs 145 pounds per cubic foot. To find the weight of the concrete block shown, you need to find its volume. The area of the base can be found as follows:
33 Theorem 11.8 is named after Bonaventura Cavalieri ( ). To see how it can be applied, consider the solids on the next slide. All three have cross sections with equal areas, B, and all three have equal heights, h. By Cavalieri s Principle, it follows that each solid has the same volume.
34 Cavalieri s Principle If two solids have the same height and the same cross-sectional area at every level, then they have the same volume.
35 Example: Find the volume of the oblique cylinder.
36 Homework: Exercises 11.6 Regular: #1 23, Honors: #1 26, 28-32
37 11.7 Volume of Pyramids and Cones Objective: Find volumes of pyramids and cones. Essential Question: How do you find the volume of a pyramid or cone?
38 Finding Volumes of Pyramids and Cones Earlier, you learned that the volume of a prism is equal to Bh, where B is the area of the base, and h is the height.
39 Theorems: Volume of a Pyramid The volume V of a pyramid 1 is, where B is the area of the base 3 and h is the height.
40 Theorems: Volume of a Cone The volume of a cone 1 1 is, 3 3 where B is the area of the base, h is the height and r is the radius of the base.
41 Example: Find the volume of the pyramid with a regular base.
42 Example: Finding the volume of a cone.
43 Example: A nautical prism is a solid piece of glass as shown. Find its volume.
44 Example: If oil is being poured into the funnel at a rate of 147 milliliters per second and flows out of the funnel at a rate of 42 milliliters per second, estimate the time it will take for the funnel to overflow. (1 ml = 1 cm 3 ).
45 Homework: Exercises 11.7 Regular: #1 25, Honors: #1 25, 29-37
46 11.8 Surface Area and Volume of Spheres Objective: Find surface areas and volume of spheres. Essential Question: How do you find the volume of a sphere?
47 Finding the Surface Area of a Sphere A circle is described as a locus of points in a plane that are a given distance from a point. A sphere is the.
48 About Spheres: The point C is called the of the sphere. A of a sphere is a segment from the center to a point on the sphere. A of a sphere is a segment whose endpoints are on the sphere.
49 More about Spheres: A is a chord that contains the center. As with all circles, the terms radius and diameter also represent distances, and the diameter is twice the radius.
50 Theorem: Surface Area of a Sphere The surface area of a sphere with radius r is.
51 Example: Find the surface area. When the radius doubles, does the surface area double?
52 More... If a plane intersects a sphere, the intersection is either a single point or a circle. If the plane contains the center of the sphere, then the intersection is a of the sphere. Every great circle of a sphere separates a sphere into two congruent halves called.
53 Example: The circumference of a great circle of a sphere is 13.8 feet. What is the surface area of the sphere?
54 Example: A baseball and its leather covering are shown. The baseball has a radius of about 1.45 inches. Estimate the amount of leather used to cover the baseball.
55 Theorem: Volume of a Sphere The volume of a sphere with radius r is. 3
56 Example: To make a steel ball bearing, a cylindrical slug is heated and pressed into a spherical shape with the same volume. Find the radius of the ball bearing to the right.
57 Homework: Exercises 11.8 Regular: #1 20, 30, 31 Honors: #1 28, 30, 31
58 11.9 Explore Similar Solids Objective: Use properties of similar solids Essential Question: If two solids are similar, what is the ratio of their surface areas and what is the ratio of their volumes?
59 Similar Solids Two solids of the same type with equal ratios of corresponding linear measures (such as heights or radii) are called similar solids.
60 Similar Solids Similar solids NOT similar solids
61 Similar Solids & Corresponding Linear Measures To compare the ratios of corresponding side or other linear lengths, write the ratios as fractions in simplest terms
62 Example: Are these solids similar?
63 Example: Are these solids similar?
64 Similar Solids and Ratios of Areas If two similar solids have a scale factor of a : b, then corresponding areas have a ratio of a 2 : b 2. This applies to lateral area, surface area, or base area
65 Similar Solids and Ratios of Volumes If two similar solids have a scale factor of a : b, then their volumes have a ratio of a 3 : b
66 Homework: Exercises 11.9 Regular: #1 18, Honors: #1 20, 25-27
67 Test
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