Geometry Solids Identify Three-Dimensional Figures Notes

Transcription

1 26 Geometry Solids Identify Three-Dimensional Figures Notes A three dimensional figure has THREE dimensions length, width, and height (or depth). Intersecting planes can form three dimensional figures or solids. A polyhedron is a solid that is composed entirely of flat surfaces called polygons. An is where two planes intersect. A is a point where three or more planes intersect. A is a flat surface. A prism is a polyhedron with two parallel, congruent faces called bases that are polygons. A pyramid is a polyhedron with one base that is any polygon. Its other faces are triangles. Prisms and pyramids are named by the shape of their bases. Name of solid: Name of solid: There are solids that are NOT polyhedrons. A cylinder is a three-dimensional figure with congruent, parallel bases that are circles connected with a curved side. A cone has only one circular base and a vertex connected by a curved side. A sphere is a set of points in space that are a given distance (radius) from a center point. Name of solid: Name of solid: Name of solid: Write the name of each solid, name of the base and record the number of lateral faces, total faces, vertices, and edges. [Shade the base(s).] NAME: NAME: Base Shape:, # of Lateral faces Faces Base Shape:, # of Lateral faces Faces Edges Vertices Edges Vertices NAME: Base Shape:, # of Lateral faces Faces NAME: Base Shape:, # of Lateral faces Faces Edges Vertices Edges Vertices NAME: Base Shape:, # of Lateral faces Faces NAME: Base Shape:, # of Lateral faces Faces Edges Vertices Edges Vertices

2 ~~ Unit 9, Page 27 ~~ Assignment Part 1: Name each polyhedron then find the faces, edges, & vertices. NAME: _ # of Vertices # of Faces # of Edges NAME: _ # of Vertices # of Faces # of Edges NAME: _ # of Vertices # of Faces # of Edges NAME: _ # of Vertices # of Faces # of Edges NAME: _ # of Vertices # of Faces # of Edges 6. NAME: _ # of Vertices # of Faces # of Edges Part 2: Use the solids to answer all parts of each problem. 1) D H A E C G B F Base shape: Name: Base names: _ and The shape of the lateral faces are Examples of 2 lateral face names: # of edges = Examples of 2 names: # of faces = # of vertices = Names:

3 ~~ Unit 9, Page 28 ~~ Assignment continued 2) B Base shape: Name: A E C Base names: _ and The shape of the lateral faces are F D Examples of 2 lateral face names: # of edges = Examples of 2 names: # of faces = # of vertices = Names: 3) E Base shape: Name: Base names: _ and A D The shape of the lateral faces are B C Examples of 2 lateral face names: # of edges = Examples of 2 names: # of faces = # of vertices = Names: Part 3: Circle the letter of the best answer for each multiple choice problem. 4) 5) 6) 7)

4 (Assignment Continued) ~~ Unit 9, Page 29 ~~ 8) In the figure below, points A, E, and H are on a 9) plane that intersects a right prism. What is the intersection of the plane with the right prism? A. A line B. A triangle C. A quadrilateral D. A pentagon 10) 11) 12) 13) 14) 15)

5 What is a cube root??? ~~ Unit 9, Page 30 ~~ 1) Write the volume (number of cubes used) and side length of each cube. A) B) C) A) Volume = Side = B) Volume = Side = C) Volume = Side = The figures above are called perfect cubes. The measure of the volume of a perfect cube is called a cubic number. List the first 6 cubic numbers. 3) Given the volume of the cubes below, find the side length of each cube. (Remember, the volume of a cube equals (length) * (width) * (height), or simply, s 3 (side cubed). A) side = V = 27 u 3 C) side = V = 343 u 3 E) side = B) side = V = 1000 u 3 V = 216 u 3 D) side = V = 729 u 3 F) What about this one. side: V = 100 u 3

6 ~~ Unit 9, Page 31 ~~ Exponents and Roots related to the AREA of SQUARES Recall that when we were working with the area of squares in the Pythagorean Unit, whenever we had the area of a square, we took the square root of it to find the length of one side of the square. For example, 16 means: What is the side length of a SQUARE with an area of 16? So when we answer, 4, we are saying that 4 is both the length and width of the square and it is multiplied by itself (4 2 ) for a square to have an area of 16 square units. A = lw, so for a square, A = s s, or s 2. The square root is s, or side length! The square root of 16 can be written in two ways: A = 25in A = 50in 2 16 or 16 NOTE: While we are not required to write the index of 2 on the square Side= Side root we MUST write the 3 with cube roots! Exponents and Roots related to the VOLUME of CUBES The cube root symbol looks like this: 3 The number written inside the cube root symbol is the volume of the cube, and the cube root symbol tells you to find the side length of that cube (s), sometimes called an edge (e). 3 For example, 8 means: What is the side length of a cube with a volume of 8? Don t forget that volume is length times width times height, and since our figures are all cubes, the length, width, and height are all the same measure! If Volume(V) = lwh, then for a cube, V = s s s, or s 3. The cube root of side 3 is side. 4) The sides of the cubes below are not whole numbers because they are not perfect cubes. So, like you did when estimating the square root of numbers that were not perfect squares, find the two whole numbers the side lengths of these cubes must be between. For example: If the volume is 40, 27 < 40 < 64, and < 40 < 64, so you would say the length of the side of a cube with volume 40 is between 3 and 4! 12 u 3 38 u u u u 3 A) < side < B) < side < C) < side < D) < side < E) < side < 5) Find the value of each cube root. If the result is not a whole number, state which two whole numbers the cube root value is between. 3 a) 27 3 b) c) 40 3 d) 64 3 e) f) g) 17 3 h) 1

7 ~~ Unit 9, Page 32 ~~ NOTES Volume Formulas: CUBIC MEASURE A cubic unit is a cube whose edge is 1 unit. Thus, a cubic inch is a cube whose sides are all 1 inch in length. cube 1 in 3 1 inch 1 inch 1 inch rectangular prism The volume of a solid is the number of cubic units it contains. Thus, a box 5 units long, 3 units wide, and 4 units high has a volume of 60 cubic units. 60 units 3 4 That is, it has a capacity or space large enough to contain 60 cubes, 1 unit on each edge. 5 3 In volume formulas, the volume is in cubic units, the unit being the same as that used for the dimensions. This means if the edge of a cube is 3 yards, its volume is (3 yd) 3 = 27 cubic yards, or 27yd 3. Calculating the Volume of Right Prisms and Cylinders To find the volume of any right prim, calculate the area of the BASE and multiply by the height. Examples V = Bh, where B is the area of the base. 1. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume: 2. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume:

8 ~~ Unit 9, Page 33 ~~ 3. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume: The formula for the volume of a cylinder is: V = πr 2 h. Where r is the of the circular face at the base of the cylinder, and h is the of the cylinder. To find the volume of the cylinder to the right, substitute the measurements into the formula above. Notice that in this figure, the diameter is given, and we need the radius. Diameter = radius, so r = Height of the cylinder = Formula: V = πr 2 h V = V = Calculate the volume of the cylinders below. Write your answers in terms of π and then round to the nearest tenth using 3.14 for π. 4) 5) 6) Radius = 4 cm 4 cm Diameter = 6 ft 10 ft Circumference = 10π u 5 cm 6 ft

9 ~~ Unit 9, Page 34 ~~ Assignment 1. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume: 2. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume: 3. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume:

10 ~~ Unit 9, Page 35 ~~ 4. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume: 5. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume: 6. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume:

11 ~~ Unit 9, Page 36 ~~ 7. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume: 8. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume: 9. Identify the polyhedron by name: Identify the base by name: _ Calculate the area of the base: Show work Calculate the volume:

12 ~~ Unit 9, Page 37 ~~ Assignment Find the volume of the cylinders below. Be sure to include cubic measurements in your answers and leave your answers in terms of π and then round to the nearest tenth. 1.5 cm 4 in 1) 2) 3) 4) 5 cm 8 in 5 in 3 in 7 cm 25 cm 5) 6) 7) Radius = 10 m Height = 4 m 4 cm 12 mm 10 cm 12 mm 8) Circumference = 8π in 9) Circumference = 5π ft Height = 25 in Height = 7 ft

13 ~~ Unit 9, Page 38 ~~ Notes - VOLUME OF 3-D FIGURES - Right Prisms and Cylinders; finding any dimension To find the volume of a right prism or cylinder, multiply the by the. Basic Volume Formula for any right prism: The Volume formula for a CYLINDER is: Example 1 Example 2 Given the Area of the Base, find the Height Answer: Height = 4 in Given the figure. find the height if the area of the base is 100m 2 and the volume is 1200m 3. Identify the base by name: _ Show work

14 ~~ Unit 9, Page 39 ~~ Examples for you to try: 3) The volume of a cylinder is 405 with a diameter of 18. Find the height of the cylinder. 4) The volume of the triangular prism is 312cm 3. Find the height is the area of the base is 52cm 2. Some of the problems in the assignment will be review, solving for the volume of the figure. Assignment: FIND THE VOLUME. Show all work. 1) 2) In terms of π: Volume = Volume to the nearest tenth. In terms of π: Volume = Volume to the nearest tenth.

15 ~~ Unit 9, Page 40 ~~ 3) The volume of a cylinder is 450 with a radius of 10. Find the height of the cylinder. Show all work. 4) Find the volume. Show all work. Find the volume of each solid. Show all work. 5) 6) (In terms of π) Volume = Find the volume to the nearest tenth. Volume (In terms of π) Volume = Find the volume to the nearest tenth. Volume

16 ~~ Unit 9, Page 41 ~~ Find the indicated dimension of each PRISM. Show all work. 7) 8) H =? W =? Volume = 1728 ft 3 Volume = 42 m 3 9) The volume of a cylinder is approximately ft 3 with a diameter of 32 ft. Find the height of the cylinder. Show all work. 10) Find the volume. Show all work. 11) Find the height. Show all work. 12) Find the height. Show all work. Area of Base: 45 cm 2 Area of Base: 58 m 2 Volume: 360 cm 3 Volume: 174 m 3

17 ~~ Unit 9, Page 42 ~~ Volume of Pyramids We will be finding the volume of rectangular and triangular right pyramids. A pyramid is considered to be right if the apex(top) is directly above the center of the base. A pyramid is Rectangular Pyramid Triangular Pyramid Remember that a prism has two congruent bases with rectangular sides. Square Pyramid Describe the following polyhedron by identifying the base and if it is a pyramid or prism. Formula for the volume of a regular pyramid: V = 1 3 Bh Find the volume. Round to the nearest tenth as needed. 1) 2)

18 ~~ Unit 9, Page 43 ~~ Find the missing dimension. 3) 4) H =? H =? Volume = ft 3 Volume = 882 ft 3 Assignment Part A) Find the volume or missing dimension of each pyramid. Show all work. 1) 2) Volume =? Volume =? 3) 4) H =? H =? Volume = 385 ft 3 Volume = 5440 ft 3

19 Part B) ~~ Unit 9, Page 44 ~~

20 ~~ Unit 9, Page 45 ~~ Part C) Find the volume of the prism, pyramid or cylinder. Identify the name first. State the formula to be used. Show and label all work. If π is used, state your answer first in terms of π, then use 3.14 for π and round to the nearest tenth. (Some dimensions are given that you won't need to use.) 1) 2) 3) 4) 5) 6) 7) 8)

21 ~~ Unit 9, Page 46 ~~ Volume of Cones NOTES: To find the volume of a cone, substitute the measurements given for the cone into the correct formula and solve. Remember, volumes are expressed using cubic units, such as in 3, ft 3, m 3, cm 3, or units 3. Volume of a CONE r Formula: V cone = 1 3 πr2 h h h r In words: The volume of a cone equals one-third the volume of a cylinder with the same radius and height! In a drawing of a cone inside a cylinder, you might see that that the triangular 1 cross-section of a cone is 1 the rectangular cross-section of the cylinder. 2 That is seeing the situation in only two dimensions. Why do you suppose the volume (which is in three dimensions) turns out to be less than 1 2 the volume of the cylinder? It actually turns out to be 1 3! NOTES Volume of Cone Formula: Find the volume of the following cones. Leave answers in terms of π, then approximate to the nearest tenth using 3.14 for π. 1) 2) 3) 9 in 8 ft 2 ft 16 in 12 cm 3 cm

22 ~~ Unit 9, Page 47 ~~ Assignment 1) 2) 3) 1 ft 42 cm 5ft 25cm 13 in 16 in Multiple Choice (Show work.) 4) 5)

23 ~~ Unit 9, Page 48 ~~ Review Find the volume of the prism, pyramid or cylinder. Identify the name first. State the formula to be used. Show and label all work. If π is used, state your answer first in terms of π, then use 3.14 for π and round to the nearest tenth. (Some dimensions are given that you won't need to use.)

24 ~~ Unit 9, Page 49 ~~ Volume of Spheres Definition Sphere the set of all points in space that are the same distances from a center point. Formula: V sphere = 4 3 πr3 Part A) For Examples 1 and 2, find the volume of each sphere. Example 1: Example 2: (Hint: What s the radius?) In terms of π Volume = In terms of π Volume = Find the volume to the nearest tenth. Volume Find the volume to the nearest tenth. Volume

25 ~~ Unit 9, Page 50 ~~ Part B) HEMISPHERES Definition HEMISPHERE a circular cross section that separates a sphere into two congruent halves. Formula V HEMISPHERE = 1 2 (4 3 πr3 ) Example 1: Find the volume of the hemisphere with a diameter of 15 km. Round to the nearest tenth. Example 2: The inside of a cereal bowl is in the shape of a hemisphere that measures 6 inches all the way across. Find the maximum amount of milk that can fit in the bowl. Round to the nearest hundredth. Part C) DETERMINING MISSING LENGTHS Example 1: The volume of a golf ball is about 13.2π cm 3. What is the radius of the golf ball to the nearest tenth? Example 2: The volume of a baseball is about cubic inches. What is the diameter of the baseball to the nearest tenth?

26 ~~ Unit 9, Page 51 ~~ Assignment Find the exact volume (leave the answer in terms of π). Then use 3.14 for π and round to the nearest tenth. Show all work.

27 ~~ Unit 9, Page 52 ~~ 10) The volume of a sphere is 288π ft 3. What is the diameter of the sphere? 11) The volume of a sphere is about cm 3. What is the approximate radius? Review: Find the volume

28 Notes ~~ Unit 9, Page 53 ~~ Volume of Composite or Combined Figures Label your work. Identify the figure name, write the formula and show all work. Name: Formula: Name: Formula: Name: Formula: Name: Formula: Combined Volume: Combined Volume:

29 ~~ Unit 9, Page 54 ~~ Assignment Label your work. Identify the figure name, write the formula and show all work Name: Formula: Name: Formula: Name: Formula: Name: Formula: Combined Volume: Combined Volume: 3. Name: Name: Formula: Formula: Combined Volume:

30 ~~ Unit 9, Page 55 ~~ Name: Formula: Name: Formula: Name: Formula: Name: Formula: Combined Volume: Combined Volume: 6. Name: Formula: Name: Formula: Combined Volume:

31 ~~ Unit 9, Page 56 ~~ Solid Geometry Word Problems NOTES Carefully read and solve the problems below. 1. Robert is using a cylindrical barrel filled with water to flatten the sod in his yard. The circular ends of the barrel have a radius of 1 foot. The barrel is 3 feet tall. How much water will the barrel hold? Find the volume formula for a CYLINDER on your reference sheet and record below. Formula: 2. If a basketball measures 24 centimeters in diameter, what volume of air will it hold? Find the volume formula for a SPHERE on your reference sheet and record below. Formula: 3. What is the volume of a sugar cone that is 2 inches in diameter and 5 inches tall? Find the volume formula for a CONE on your reference sheet and record below. Formula:

32 ~~ Unit 9, Page 57 ~~ Practice Find the volume of each solid. Show all work. 1) Approximately how much air would be needed to fill a dozen soccer balls with a radius of 14cm? Round to the nearest hundredth. 2) Find the volume of the following figure if the diameter is 4.5 in and the height of the cylinder is 2.5 in. Round to the nearest tenth. 3) The diameter of the earth is approximately 7,926 miles. The diameter of the moon is approximately 2,159 miles. Approximately how many moons would fit inside the earth? 4) Find the radius of a sphere with a volume of 1,767.1 m 3. Round to the nearest tenth. 5) Find the radius of a hemisphere with a volume of 2,712.3 in 3. Round to the nearest tenth.

33 ~~ Unit 9, Page 58 ~~ Review 1. Find the difference between the volumes of the two objects below. 4 in 10 in 4 in 4 in 8 in 4 in 10 in 7 in 6 in 2. Find the volume of the compound figure below. 4 in 10 in 8 in Directions: Find the volume of the following figures and situations. 3) 4) Volume = Volume =

34 ~~ Unit 9, Page 59 ~~ 5) Find the volume to the nearest tenth. Volume 6) Find the volume of a cylindrical cake that is 5 in. tall with a radius of 7.5 in. Happy Birthday! 7) A standard men s basketball has a circumference of about 29.5 inches. What is the volume of the basketball to the nearest hundredth? (hint: find the diameter first.) 8) A cylindrical container is used to hold dog food. Its volume is approximately ft 3 and has a radius of 2 ft. What is the height of the container to the nearest foot? lucky Dog Food stay fresh container

35 ~~ Unit 9, Page 60 ~~ 9) A globe in a brass stand has an approximate volume of 33, in 3, what is its radius length? 10) Find the volume. 11) Find the volume, to the nearest tenth, of a 4 ft by 2 ft by 3 ft rectangular prism with a cylindrical hole, radius 6 in., through the center. 12) Marge has a cylindrical tin of popcorn that is 18 in. tall and has a radius of 4 in. She wants to use the tin for something else and needs to empty the popcorn into a box. The box is 8 in. long, 8 in. wide and 14 in. tall. Will the popcorn fit in the box? Explain. Tin Can

36 ~~ Unit 9, Page 61 ~~ 13) Spaceship Earth at Epcot Center in Florida is a 180-foot geosphere. Find the volume by assuming it is a sphere with a diameter of 180 feet. 14) The volume of the following soup can is 22π in 3, and has a height of 5.5 in. What is the radius of the soup can? 15) Based on the following drawing, if the top funnel was filled with water and then emptied into the bottom cone, what fraction of the bottom cone would be filled with water? Explain.

37 ~~ Unit 9, Page 62 ~~ 16) A cylinder is 9 inches high. The circumference of the base is 12π inches. Find the volume. 17) The height of a cylinder is 10 and the area of a base is 36π square units. What is the volume in cubic units? 18) A can of soup contains about 553 cubic centimeters of soup. The height of the can is 11 cm. What is the approximate diameter of the can to the nearest centimeter? 19) A scented candle is in the shape of a cylinder, with a radius of 4cm and a height of 12cm. Find the volume (leave in terms of π).

38 ~~ Unit 9, Page 63 ~~ 20) A cylindrical cake takes up 32π cubic inches. The diameter of the cake is 8 inches, what is the height of the cake? Happy Birthday! 21) Nate uses a cube shaped bead with side lengths measuring 6mm. Each bead has a circular hole in the middle. The diameter of the circular hole is 3mm. Find the volume of the bead. In terms of π, Volume = Now find the volume to the nearest tenth. Volume 22) If the volume of a cube is 729 cubic feet, then what is the length of one edge of the cube? e 23) Multiple Choice Find the volume of concrete used to construct the ramp. A) 30 ft 3 C) 66 ft 3 B) 36 ft 3 D) 96 ft 3 2 ft 3 ft 10 ft 6 ft

39 ~~ Unit 9, Page 64 ~~ 24) The volume of a cylinder is about 1632 in 3. The height of the cylinder is 24in. What is the area of the base? 25) Tanya uses a cube shaped bead with side lengths measuring 12mm. Each bead has a circular hole in the middle. The diameter of the circular hole is 2mm. Find the volume of the bead. [Hint: use the diagram.] 26) A chocolate bar is in the shape of a trapezoidal prism as shown below. Find the volume of the chocolate bar. 5 cm 7 cm 3 cm 3 cm 3 cm 27) Multiple Choice 28) Multiple Choice Find the maximum amount of water that can fill the trough shown. A) 20.5 ft 3 B) 24.5 ft 3 A) 25.1 in 3 C) 48 ft 3 B) in 3 D) 49 ft 3 C) in ft What is the volume of a cylinder with a radius of 8 inches and a height of 1 foot? Round answer to the nearest tenth. D) in 3 10 ft

40 ~~ Unit 9, Page 65 ~~ 29) Tennis balls with a diameter of 3 inches are sold in cans of three. The can is in the shape of a cylinder. What is the volume of the space NOT occupied by the tennis balls? Assume the tennis balls touch the can on the sides, top and bottom. Round your answer to the nearest tenth. 3 in 30) 31) In terms of π: Volume = Find the volume to the nearest tenth. In terms of π: Volume = Find the volume to the nearest tenth. Volume Volume 32) Multiple Choice The diagram represents a tower. The tower is in the shape of a cone on top of a cylinder. Which measurement is closest to the total volume of the tower? A) 2,200 cubic meters B) 2,600 cubic meters C) 9,400 cubic meter D) 10,500 cubic meters

41 ~~ Unit 9, Page 66 ~~ The volume of a cone is 405 in 3 with a diameter of 18in. Find the height of the cone. 36. An ice cream shop designs a new ice cream cone. He wants the volume to be about 240cm 3. The cone is 14cm tall. What is its radius to the nearest whole number?