# 3 Dimensional Solids. Table of Contents. 3 Dimensional Solids Nets Volume Prisms and Cylinders Pyramids, Cones & Spheres

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1 Table of Contents 3 Dimensional Solids Nets Volume Prisms and Cylinders Pyramids, Cones & Spheres Surface Area Prisms Pyramids Cylinders Spheres More Practice/ Review 3 Dimensional Solids Polyhedron A 3 D figure whose faces are all polygons Sort the figures into the appropriate side. Polyhedron Not Polyhedron 1

2 3 Dimensional Solids Categories & Characteristics of 3 D Solids: Prisms 1. Have 2 congruent, polygon bases which are parallel to one another 2. Sides are rectangular (parallelograms) 3. Named by the shape of their base Pyramids 1. Have 1 polygon base with a vertex opposite it 2. Sides are triangular 3. Named by the shape of their base 3 Dimensional Solids Categories & Characteristics of 3 D Solids: Cylinders 1. Have 2 congruent, circular bases which are parallel to one another 2. Sides are curved Cones 1. Have 1 circular bases with a vertex opposite it 2. Sides are curved 3 Dimensional Solids Vocabulary Words for 3 D Solids: Polyhedron : A 3 D figure whose faces are all polygons (Prisms & Pyramids) Face : Edge : Flat surface of a Polyhedron Line segment formed where 2 faces meet Vertex (Vertices): Point where 3 or more faces/edges meet 2

3 1 Name the figure. A B C D E F Rectangular Prism Triangular Pyramid Hexagonal Prism Rectangular Pyramid Cylinder Cone 2 Name the figure A B C D E F Rectangular Pyramid Triangular Prism Octagonal Prism Circular Pyramid Cylinder Cone 3 Name the figure A B C D E F Rectangular Pyramid Triangular Pyramid Triangular Prism Hexagonal Pyramid Cylinder Cone 3

4 4 Name the figure A B C D E F Rectangular Prism Triangular Prism Square Prism Rectangular Pyramid Cylinder Cone 5 Name the figure A B C D E F Rectangular Prism Triangular Pyramid Circular Prism Circular Pyramid Cylinder Cone For each figure, the number of faces, vertices and edges is below. Can you figure out a relationship between the number of faces, vertices and edges of 3 Dimensional Figures? Cube Name Faces Vertices Edges Rectangular Prism Triangular Prism Triangular Pyramid Square Pyramid Pentagonal Pyramid Octagonal Prism 4

5 Euler's Formula F + V 2 = E The number of edges is 2 less than the sum of the faces and vertices. 6 How many faces does a pentagonal prism have? 7 How many edges does a rectangular pyramid have? 5

6 8 How many vertices does a triangular prism have? Nets Nets Nets are two dimensional drawings that represent the surface area of three dimensional shapes. There is more than one way to draw a net for a cube, however not all nets can be folded into a cube... 6

7 How many different arrangements can be folded into a cube? Nets for prisms will have rectangular faces and two bases for which the shape is named. Notice the two triangles are opposite from one another (bases). Nets for pyramids will have triangular faces and ONE face (base) for which the shape is named. Square pyramid 7

8 9 Name this polyhedron. A B C D Hexagonal prism Pentagonal pyramid Pentagonal prism Rectangular prism 10 Name the polyhedron. A B C D Hexagonal prism Hexagonal pyramid Triangular prism Triangular pyramid 11 Name the figure. A B C D Triangular Prism Triangular Pyramid Rectangular Prism Rectanglular Pyramid 8

9 12 Name the figure. A B C D Circular cone Rectangular cone Cylinder Circular pyramid 13 Name the figure. A B C D Triangular Prism Triangular Pyramid Pentagonal Prism Square Prism Volume 9

10 Volume Volume The amount of space occupied by a 3 D Figure The number of cubic units needed to FILL a 3 D Figure (layering) Label Units 3 or cubic units Volume of Prisms & Cylinders Volume Volume of Prisms & Cylinders: Area of Base x Height A = Bh Area Formulas: (used to find area of base) Rectangle = lw or bh Triangle = bh or 2 (bh) Circle = r 2 10

11 Find the Volume. 8 m 2 m 5 m Find the Volume. 9 yd 10 yd 14 Find the Volume. 4 in in 1 1 in 2 11

12 15 Find the volume of a rectangular prism with length 2 cm, width 3.3 cm and height 5.1 cm. 16 Which is a possible length, width and height for a rectangular prism whose volume = 18 cm 3 A 1 x 2 x 18 B 6 x 3 x 3 C 2 x 3 x 3 D 3 x 3 x 3 17 Find the volume. 47 ft 50 ft 42 ft 21 ft 12

13 18 A box shaped refrigerator measures 12 by 10 by 7 on the outside. All six sides of the refrigerator are 1 unit thick. What is the inside volume of the refrigerator in cubic units? HINT: You may want to draw a picture! 19 Find the volume. 10 m 6 m 20 Which circular glass holds more water? A B Glass A having a 7.5 cm diameter and standing 12 cm high Glass B having a 4 cm radius and a height of 11.5 cm 13

14 21 What is the volume of the largest cylinder that can be placed into a cube that measures 10 feet on an edge? 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 for π. Volume of Pyramids, Cones & Spheres 14

15 Given the same diameter and height for each figure, drag them to arrange in order of smallest to largest volume. How many filled cones do you think it would take to fill the cylinder? How many filled spheres do you think it would take to fill the cylinder? Reminder! Volume of a Cylinder Volume = Area of Base x Height V = Bh V = r 2 x h Volume of a Cone A cone is 1/3 the volume of a cylinder with the same base area (B) and height (h). V = 1/3 (Volume of a Cylinder) V = 1/3 (Area of Base x Height) V = (Area of Base x Height) 3 V = ( r 2 x h) 3 15

16 Volume of a Sphere A sphere is 2/3 the volume of a cylinder with the same base area (B) and height (h). V = 2/3 (Volume of Cylinder) V = 2/3 (Area of Base x Height) V = 2/3 ( r 2 x h) OR V = 4/3 r 3 How much ice cream can a Friendly s Waffle cone hold if it has a diameter of 6 in and its height is 10 in? (Just Ice Cream within Cone. Not on Top) 23 Find the volume. 9 in 4 in 16

17 24 Find the Volume 8 cm 5 cm If the radius of a sphere is 5.5 cm, what is its volume? 25 What is the volume of a sphere with a radius of 8 ft? 17

18 26 What is the volume of a sphere with a diameter of 4.25 in? Reminder! Volume of a Prism Volume = Area of Base x Height 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). V = 1/3 (Volume of Prism) V = 1/3 (Area of Base x Height) V = (Area of Base x Height) 3 18

19 Pyramids are named by the shape of their base.. The volume is a pyramid is 1/3 the volume of a prism with the same base area(b) and height (h). 1 V = Bh 3 1 V = Bh 3 =5 m side length = 4 m 27 Find the Volume of a triangular pyramid with base edges of 8 in, base height of 4 in and a pyramid height of 10 in. 10 in 4 in 8 in 28 Find the volume cm 7 cm 8 cm 19

20 Surface Area Surface Area of Prisms Surface Area 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 many surfaces are there? How do you find the area of each surface? 8 in 6 in 3 in 20

21 Surface Area 6 in 8 in 3 in Bottom TopLeft Right Front Back SUM x 8 x 3 x in 6 in 6 in 8 in 3 in 8 in 3 in 8 in 3 in Arrangement of Unit Cubes Find all the possible ways that 24 unit cubes can be arranged into a rectangular prism. Give each arrangement's dimensions, volume, and surface area. Length Width Height Volume Surface Area Which arrangement of 24 cubes has the least surface area? Which has the most? 21

22 Arrangement of Unit Cubes Find all the possible ways that 64 unit cubes can be arranged into a rectangular prism. Give each arrangement's dimensions, volume, and surface area. Length Width Height Volume Surface Area Which arrangement of 64 cubes has the least surface area? Which has the most? 29 Which arrangement of 27 cubes has the least surface area? A 1 x 1 x 27 B 3 x 3 x 3 C 9 x 3 x 1 22

23 30 Which arrangement of 12 cubes has the least surface area? A 2 x 2 x 3 B 4 x 3 x 1 C 2 x 6 x 1 D 1 x 1 x Which arrangement of 25 cubes has the greatest surface area? A 1 x 1 x 25 B 1 x 5 x 5 32 Which arrangement of 48 cubes has the least surface area? A 4 x 12 x 1 B 2 x 2 x 12 C 1 x 1 x 48 D 3 x 8 x 2 E 1 x 2 x 24 F 4 x 3 x 4 G 1 x 8 x 6 H 4 x 2 x 6 I 3 x 16 x 1 23

24 Find the surface area of a rectangular shoe box that has a length of 12 inches, a width of 6 inches and a height of 5 inches. 5 in 12 in Top/Bottom Front/Back Left/Right Surface Area 6 in Name the figure. Find the figure's surface area. 7 m 3 m 5 m 33 How many faces does the figure have? 6 m 2 m 4 m 24

25 34 How many area problems must you complete when finding the surface area? 6 m 2 m 4 m 35 What is the area of the top or bottom face? 6 m 2 m 4 m 36 What is the area of the left or right face? 6 m 2 m 4 m 25

26 37 What is the area of the front or back face? 6 m 2 m 4 m 38 What is the surface area of the figure? 6 m 2 m 4 m Find the Surface Area. 1. Draw and label ALL faces 2. Find the correct dimensions for each face 3. Calculate the AREA of EACH face 4. Find the SUM of ALL faces 5. Label Answer 5 yd 5 yd 4 yd 6 yd 9 yd go on to see steps 26

27 5 yd 5 yd 5 yd 5 yd 6 yd 4 yd 9 yd 5 yd 6 yd 4 yd Triangles Bottom Rectangle 4 9 x 6 x 6 24 / 2 = x 2 24 Total Left/Right Rectangles 6 yd 4 yd 9 yd 5 yd 9 yd (Same size since isosceles) 5 x 9 45 x 2 90 Surface Area yd 2 Find the Surface Area. 1. Draw and label ALL faces 2. Find the correct dimensions for each face 3. Calculate the AREA of EACH face 4. Find the SUM of ALL faces 5. Label Answer 9 cm go on to see steps 9 cm 6 cm 11 cm 9 cm 9 cm 9 cm 9 cm 6 cm 11 cm 9 cm 6 cm 11 cm 9 cm Triangles Rectangles 9 (All 3 are the same size X6 since triangles are equilateral) 54 / 2 = 27 9 x 2 x x cm Surface Area cm 2 27

28 39 Find the surface area of the shape below. 1. Draw and label ALL faces 2. Find the correct dimensions for each face 3. Calculate the AREA of EACH face 4. Find the SUM of ALL faces 5. Label Answer 47 ft 21 ft 50 ft 42 ft Find the Surface Area. 7 cm 3 cm 4 cm 5 cm 15 cm 6 cm 7 cm 3 cm 4 cm 5 cm 15 cm 6 cm Trapezoids Bottom Rectangle Top Rectangle Side Rectangles Surface Area 28

29 40 Find the surface area of the shape below. 8 cm 6 cm 9 cm 10 cm 41 Find the surface area of the shape below. 10 cm 2 cm 6 cm 6 cm 10 cm Surface Area of Pyramids 29

30 Surface Area of Pyramids What is a pyramid? Polyhedron with one base and triangular faces that meet at a vertex How do you find Surface Area? Sum of the areas of all the surfaces of a 3 D Figure Find the Surface Area cm go on to see steps 8 cm 8 cm Find the Surface Area cm 17.5 cm 17.5 cm 8 cm 8 cm 8 cm 8 cm Bottom Rectangle A = l x w A = 8 x 8 A = 64 8 cm Front/Back Triangles 8 cm Left/Right Triangles Surface Area cm2 30

31 Find the surface area of a square pyramid with base edge of 4 inches and triangle height of 3 inches. 3 in Base Triangles Surface Area 4 in Find the surface area. Be sure to look at the base to see if it is an equilateral or isosceles triangle (making all or two of the side triangles equivalent!). Base Triangles (all equal) 4 in 6 in 4 in 3.5 in 4 in Surface Area 42 Which has a greater Surface Area, a square pyramid with a base edge of 8 in and a height of 4 in or a cube with an edge of 5 in? A B Square Pyramid Cube 31

32 43 Find the Surface Area of a triangular pyramid with base edges of 8 in, base height of 4 in and a slant height of 10 in. 8 in 10 in 8 in 6.9 in 8 in 44 Find the Surface Area. 12 m 11 m 9 m 6.7 m 9 m Surface Area of Cylinders 32

33 How would you find the surface area of a cylinder? Notice the length of the rectangle is actually the circumference of the circular base. Steps 1. Find the area of the 2 circular bases. 2. Find the area of the curved surface (actually, a rectangle). 3. Add the two areas. 4. Label answer. Radius Radius Curved Side = Circumference of Circular Base H E I G H T H E I G H T 33

34 Radius Radius Curved Side = Circumference of Circular Base H E I G H T H E I G H T 2 r 2 + dh or 2 r rh r 2 Area of Circles = 2 ( ) Area of Curved Surface = Circumference Height d Height Find the surface area of a cylinder whose height is 14 inches and whose base has a diameter of 16 inches. 16 in Area of Circles = 2 ( r 2 ) 14 in Area of Curved Surface = Circumference x Height = d x h Surface Area = 45 Find the surface area of a cylinder whose height is 8 inches and whose base has a diameter of 6 inches. 34

35 46 Find the surface area of a cylinder whose height is 14 inches and whose base has a diameter of 16 inches. 47 How much material is needed to make a cylindrical orange juice can that is 15 cm high and has a diameter of 10 cm? 48 Find the surface area of a cylinder with a height of 14 inches and a base radius of 8 inches. 35

36 49 A cylindrical feed tank on a farm needs to be painted. The tank has a diameter 7.5 feet and a height of 11 ft. One gallon of paint covers 325 square feet. Do you have enough paint? Explain. Yes No Surface Area of Spheres A sphere is the set of all points that are the same distance from the center point. Like a circle, a sphere has a radius and a diameter. You will see that like a circle, the formula for surface area of a sphere also includes pi. Surface Area of a Sphere Radius 36

37 If the diameter of the Earth is 12,742 km, what is its surface area? 12,742 km Try This: Find the surface area of a tennis ball whose diameter is 2.7 inches. 2.7 in 50 Find the surface area of a softball with a diameter 3.8 inches. 37

38 51 How much leather is needed to make a basketball with a radius of 4.7 inches? 52 How much rubber is needed to make 6 racquetballs with a diameter of 5.7 inches? More Practice / Review 38

39 53 Find the volume. 22 mm 8 mm 15 mm 54 Find the volume of a rectangular pyramid with a base length of 2.7 meters and a base width of 1.3 meters, and the height of the pyramid is 2.4 meters. HINT: Drawing a diagram will help! 55 Find the volume of a square pyramid with base edge of 4 inches and pyramid height of 3 inches. 39

40 56 Find the Volume 11 m 9 m 12 m 6 m 9 m 57 Find the Volume 21 ft 14 ft 58 Find the Volume 8 in 6.9 in 40

41 59 Find the Volume 9 ft 4 ft 8 ft 7 ft 60 A cone 20 cm in diameter and 14 cm high was used to fill a cubical planter, 25 cm per edge, with soil. How many cones ful of soil were needed to fill the planter? 20 cm 14 cm 25 cm 61 Find the surface area of the cylinder. Radius = 6 cm and Height = 7 cm 41

42 62 Find the Surface Area. 11 cm 11 cm 12 cm 63 Find the Surface Area. 9 in 9 in 2 in 7 in 8 in 64 Find the Volume. 9 in 9 in 2 in 7 in 8 in 42

43 65 A rectangular storage box is 12 in wide, 15 in long and 9 in high. How many square inches of colored paper are needed to cover the surface area of the box? 66Find the volume. 70 m 80 m 40 m 67Find the surface area of a square pyramid with a base length of 4 inches and slant height of 5 inches. 43

44 68 A teacher made 2 pair of foam dice to use in math games. Each cube measured 10 in on each side. How many square inches of fabric were needed to cover the 2 cubes? Define the following terms: Surface Area Volume Name a 3 D figure that has 6 rectangular faces. 44

45 Name a 3 D Figure that is not a polyhedron. 45

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