February 07, Dimensional Geometry Notebook.notebook. Glossary & Standards. Prisms and Cylinders. Return to Table of Contents

Transcription

1 Prisms and Cylinders Glossary & Standards Return to Table of Contents 1

2 Polyhedrons 3-Dimensional Solids A 3-D figure whose faces are all polygons Sort the figures into the appropriate side. 2. Sides are rectangular (parallelograms) click to reveal Polyhedron Not Polyhedron click to reveal 3-Dimensional Solids click to reveal Polyhedron A 3-D figure whose faces are all polygons (Prisms & Pyramids) Face Flat surface of a Polyhedron Edge Line segment formed where 2 faces meet Cones click to reveal Vertex Point where 3 or more faces/edges meet (pl. Vertices) 2

3 Name the figure. Name the figure. Triangular Pyramid Hexagonal Prism Cylinder Octagonal Prism Cylinder Name the figure. Name the figure. Hexagonal Pyramid Cylinder Cylinder 3

4 Name the figure. Cylinder For each figure, find the number of faces, vertices and edges. Can you figure out a relationship between the number of faces, vertices and edges of 3-Dimensional Figures? Faces, Vertices and Edges Euler's Formula Euler's Formula: E + 2 = F + V The sum click of to the reveal edges and 2 is equal to the sum of the faces and vertices. 6 4

5 7 8 Cross Sections 3-Dimensional figures can be cut by planes. When you cut a 3-D figure by a plane, the result is a 2-D figure, called a cross section. Cross Sections of Three-Dimensional Return to Table of Contents 5

6 Cross Sections Cross Sections Can you describe a vertical cross-section of a cone? Cross Sections Cross Sections 6

7 Which figure has the same horizontal and vertical cross- 7

8 14 Misha has a cube and a right-square pyramid that are made of clay. She placed both clay figures on a flat surface. Select each choice that identifies the two-dimensional plane sections that could result from a vertical or horizontal slice through each clay figure. A Cube cross section is a Triangle B Cube cross section is a Square C Cube cross section is a Rectangle (not a square) D Right-Square Pyramid cross section is a Triangle E Right-Square Pyramid cross section is a Square F Right-Square Pyramid cross section is a Rectangle (not a square) From PARCC EOY sample test calculator #11 Volume click to reveal Units click to reveal Return to Table of Contents 8

9 Click the link below for the activity Volume of Prisms & Cylinders Return to Table of Contents Volume of Prisms & Cylinders: click Area of Base x Height Rectangle = lw click or bh click click 9

10 15 Find the volume. Use 3.14 as your value of π

11 Teachers: Use this Mathematical Practice Pull Tab for the next 3 SMART Response slides. 11

12 21 π 22 A circular garden has a diameter of 20 feet and is surrounded by a concrete border that has a width of three feet and a depth of 6 inches. What is the volume of concrete in the path? Use 3.14 as your value of π. 23 Teachers: Use this Mathematical Practice Pull Tab for the next SMART Response slide. standing 12 cm high Glass B having a 4 cm radius and a height 12

13 volume. Return to Table of Contents How many filled cones How many filled spheres The Volume of a Cone is 1/3 the volume of a cylinder with the same base area (B) and height (h). click to reveal 13

14 The Volume of a Sphere is 2/3 the volume of a cylinder with the same base area (B) and height (h). click to reveal Find the volume. 14

15 26 27 The Volume of a Pyramid is 1/3 the volume of a prism with the same base area (B) and height (h). (Area of Base x Height) 3 1 click to reveal 3 (Area of Base x Height) 15

16 Example: Find the volume of the pyramid shown below. V = Bh V = Surface Area Return to Table of Contents 16

17 Surface Area is the sum of the areas of all outside surfaces of a 3-D figure. To find surface area, you must find the area of each surface of the figure then add them together. What type of figure is pictured? How do you find the area of each Return to Table of Contents Net Another way that you can visualize the entire surface and calculate the surface area is to create the net of your solid by unfolding it. Below is a the net of a rectangular prism. 8 6 x 3 x 6 x 6 Back Left Bottom Right Top Front 17

18 Arrangement of Unit Cubes You can also calculate the surface area of our last example by drawing the net, calculating the areas, and adding them together. Click the link below for the activity 48 in 2 18 in 2 24 in 2 18 in 2 24 in 2 48 in 2 Total Surface Area: = 180 in 2 30 surface area? Teachers: Use this Mathematical Practice Pull Tab for the next 4 SMART Response slides. 18

19 31 surface area? 32 Which arrangement of 25 cubes has the 33 Which arrangement of 48 cubes has the least surface area? Surface Area Front/Back 19

20 Name the figure. Find the figure's surface area. Surface Area 34 How many faces does the figure have? Click to reveal the prism's net, if needed 35 the surface area? 36 20

21 37 What is the area of the left or right face? 38 What is the area of the front or back face? 39 What is the surface area of the figure? 1. Draw and label ALL faces; use the net, if it's helpful 4. Find the SUM of ALL faces go on to see steps Click to reveal the prism's net, if needed 21

22 Find the Surface Area Using the Net Bottom Rectangle 9 (Same size since isosceles) (Same size since isosceles) 1. Draw and label ALL faces; use the net if it's helpful 4. Find the SUM of ALL faces A = 7.8 x 9 2 A = 35.1 cm 2 x cm 2 click to reveal Rectangles A = 9(11) = 99 cm 2 A = 99 x 3 = 297 cm 2 click to reveal 22

23 40 Find the surface area of the shape below. 1. Draw and label ALL faces; use the net if it's helpful 4. Find the SUM of ALL faces A = 9(11) = 99 cm 2 click to reveal A = 99 x 3 = 297 cm 2 click to reveal Find the Surface Area. Bottom Rectangle 18 x 4 x 2 23

24 41 42 Find the surface area of the shape below. Return to Table of Contents 24

25 go on to see steps Rectangle A = 1 2 bh(2) A = 1 2 (8)(17.4)(2) A = cm 2 A = 1 2 bh(2) A = 1 2 (7)(17.5)(2) A = cm 2 Find the surface area of a square pyramid with base edge of 4 inches and triangle height of 3 inches. isosceles triangle (making all or two of the side triangles Surface Area x Surface Area 25

26 43 Which has a greater Surface Area, a square pyramid with a Surface Area of Cylinders 9 m 9 m Return to Table of Contents 26

27 How would you find the surface area of a cylinder? Original cylinder Middle step to get to the net πr 2 Area of Curved Surface = Circumference Height Net of a cylinder π π π π 27

28 46 Area of Circles = 2 (πr 2 ) = 2 (π8 2 ) = 2 (64π) = 128π = in 2 = π d Height = π(16)(14) π = in 47 Find the surface area of a cylinder whose height is 14 inches and whose base has a diameter of 20 inches. Use 3.14 as your value of π. 48 How much material is needed to make a cylindrical orange juice can that is 15 cm high and has a diameter of 10 cm? Use 3.14 as your value of π. 28

29 49 inches and a base radius of 8 inches. Use 3.14 as your value of π. 50 value of π. Use 3.14 as your Surface Area of Spheres click to reveal Return to Table of Contents 29

30 Use 3.14 as your value of π. click to reveal 51 Use 3.14 as your value of π. 52 Use 3.14 as your value of π. 30

31 53 Use 3.14 as your value of π. Return to Table of Contents HINT: Drawing a diagram will help! 31

32 56 57 Find the Volume. 58 Find the Volume. Use 3.14 as your value of π. 59 Find the Volume. Use 3.14 as your value of π. 8 in 6.9 in 32

33 60 61 π 62 π 63 33

34 Find the surface area of a square pyramid with a base 34

35 68 69 Cone Glossary & Standards A 3-D solid that has 1 circular base with a vertex opposite it. The sides are curved. Return to Contents 35

36 Cross Section The shape formed when cutting straight through an object. Cylinder A solid that has 2 congruent, circular bases which are parallel to one another. The side joining the 2 circular bases is a curved rectangle. Triangle Trapezoid Face Flat surface of a Polyhedron. Edge Line segment formed where 2 faces meet. 1 face 1 face 1 face 1 edge 1 edge 1 edge 36

37 Euler's Formula The sum of the edges and 2 is equal to the sum of the faces and vertices. E + 2 = F + V Net A 2-D pattern of a 3-D solid that can be folded to form the figure. An unfolded geometric solid. pyramid: vertices = 4 faces = 4 E + 2 = F + V E + 2 = E + 2 = 8 E = 6 Solid Net Solid Net Solid Net Polyhedron A 3-D figure whose faces are all polygons. A Polyhedron has NO curved surfaces. Plural: Polyhedra Prism A polyhedron that has 2 congruent, polygon bases which are parallel to one another. Remaining sides are rectangular(parallelograms). Named by the shape of the base. Yes, Polyhedron Yes, Polyhedron No, not a polyhedron Triangular Prism Hexagonal Prism Octagonal Prism 37

38 Pyramid A polyhedron that has 1 polygon base with a vertex opposite it. Remaining sides are triangular. Named by the shape of their base Surface Area The sum of the areas of all outside surfaces of a 3-D figure. Pentagonal Pyramid Rectangular Pyramid Triangular Pyramid 1. Find the area of each surface of the figure 2. Add all of the areas together u u 2 30 u u SA = 84 units 2 Vertex Volume Point where 3 or more faces/edges meet Plural: Vertices The amount of space occupied by a 3-D Figure. The number of cubic units needed 1 vertex Label: 1 vertex 1 vertex cylinder filled halfway with water V =? V =? Prism filled with water Units 3 or cubic units 38

39 Volume of a Cone A cone is 1/3 the volume of a cylinder with the same base area (B = πr 2 ) and height (h). Volume of a Cylinder Found by multiplying the Area of the base (B) and the height (h). or πr 2 h h V = r πr 2 h 6 4 V = 1 / 3 π(4) 2 (6) V = 32π units 3 V = u 3 Since your base is always a circle, your volume formula for a cylinder is V = Bh V = πr 2 h r V = πr 2 h h 4 10 V = π(4) 2 (10) V = 160π units 3 V = u 3 Volume of a Prism Found by multiplying the Area of the base (B) and the height (h). V = Bh Volume of a Pyramid A pyramid is 1/3 the volume of a prism with the same base area (B) and height (h). The shape of your base matches the name of the prism Rectangular Prism V = Bh V = (lw)h Triangular Prism V = Bh 1 h The shape of your base matches the name of the pyramid Rectangular V = 1 / 3 Bh V = 1 / 3 (lw)h Triangular Pyramid V = 1 / 3 Bh V = ( 1 / 2 b h )h pyramid h b h pyramid 39

40 Volume of a Sphere A sphere is 2/3 the volume of a cylinder with the same base area (B = πr 2 ) and height (h = d = 2r). MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. h = d h = 2r πr 2 h 2/3 πr 2 (2r) π π V = 288π u 3 V = u 3 MP5 Use appropriate tools strategically. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning. Click on each standard to bring you to an example of how to meet this standard within the unit. 40