CHAPTER 12. Extending Surface Area and Volume

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1 CHAPTER 12 Extending Surface Area and Volume 0

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3 Learning Targets Students will be able to draw isometric views of three-dimensional figures. Students will be able to investigate cross-sections of three-dimensional figures. Section 12.1 Notes: Representations of Three-dimensional Figures Vocabulary, Example Types Definitions, Pictures and Examples Cross Section Types of Slice From the above figures, Identify the type of shape resulting from a vertical, angled and horizontal cross section of a cylinder. Example 1: Example 2: Example 3: 2

4 Describe the shape resulting from a vertical, angled, and horizonal cross section of a square pyramid. You Try! Vertical: Angled: Horizontal: Examples of cross sections. Example 4: BAKERY A customer ordered a two-layer sheet cake. Determine the shape of each cross section of the cake below. Example 5 :Determine the shape of the cross section shown. a. b. You Try! Describe the shape resulting from each cross section. a. 3

5 b. c. Summary Describe the cross section. 4

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7 Geometry Section 12.1 Worksheet Describe each cross section. Name: A standard basketball is shaped like a sphere. a. Draw a basketball with a vertical, angled, and horizontal slice. b. Describe the cross section made by each slice. 6

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9 Learning Targets Students will be able to find areas of regular polygons. Find areas of composite figures. Section 11.4 Notes: Areas of Regular Polygons and Composite Figures Vocabulary, Example Types Definitions, Pictures and Examples Circumscribed Inscribed Regular Polygon Apothem Central Angle Identify the center, radius, apothem and central angle. Example 1: In the figure, pentagon PQRST is inscribed in circle X. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. 8

10 Example 2: The top of the table shown is a regular hexagon with a side length of 3 feet and an apothem of 1.7 feet. What is the area of the tabletop to the nearest tenth? Area of a Regular polygon Example 3: The perimeter of regular polygon is 24 and apothem is 3. Find the polygon s area. Find the area of the regular polygons. Example 4: If a regular octagon with 3cm sides has an area of 60 square cm, what is the length of the apothem? Example 5: Find the area of the regular pentagon. Number of Sides: Central Angle: Side Length: Radius: Apothem: Perimeter: Area: 9

11 Example 6: Find the area of the regular hexagon. Round to the nearest tenth. Number of Sides: Central Angle: Side Length: Radius: Apothem: Perimeter: Area: Example 7: Find the area of the regular pentagon. Round to the nearest tenth. Number of Sides: Central Angle: Side Length: Radius: Apothem: Perimeter: Area: You Try! 1. Find the area of the regular triangle. Round to the nearest tenth. Number of Sides: Central Angle: Side Length: Radius: Apothem: Perimeter: Area: 10

12 2. Find the area of the regular triangle. Round to the nearest tenth. Number of Sides: Central Angle: Side Length: Radius: Apothem: Perimeter: Area: Summary Composite Figure 11

13 Example 8: The dimensions of an irregularly shaped pool are shown. What is the area of the surface of the pool? Example 9: Find the area of the figure in square feet. Round to the nearest tenth if necessary Example 10: Find the area of the shaded figure. Example 11: Cara wants to wallpaper one wall of her family room. She has a fireplace in the center of the wall. Find the area of the wall around the fireplace. 12

14 1. Detective LeRue must investigate a crime committed at the local park. How many square meters of ground must he cover when looking for clues? You Try! 2. Find the area and perimeter. 3. Find the area and perimeter. JoAnn wants to lay 12 x 12 tile on her bathroom floor. a. Find the area of the bathroom floor in her apartment floor plan. b. If the tile comes in boxes of 15 and JoAnn buys no extra tiles, how many boxes will she need? Summary! 13

15 Geometry Section 11.4 Regular Polygon Area Worksheet Find the area of each regular polygon. Round to the nearest tenth Number of Sides: Central Angle: Side Length: Radius: Apothem: Perimeter: Area: Name: Number of Sides: Central Angle: Side Length: Radius: Apothem: Perimeter: Area: Number of Sides: Central Angle: Side Length: Radius: Apothem: Perimeter: Area: Number of Sides: Central Angle: Side Length: Radius: Apothem: Perimeter: Area: 5. Number of Sides: 6. Central Angle: Side Length: Radius: Apothem: Perimeter: Area: Number of Sides: Central Angle: Side Length: Radius: Apothem: Perimeter: Area: 14

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17 Geometry Section 11.4 Composite Figure Area Worksheet Find the area of each figure. Round to the nearest tenth if necessary Name: MINIATURE GOLF The plan for a miniature golf hole is shown below. The right angle in the drawing is a central angle. What is the area of the playing surface? Round your answer to the nearest hundredth of a square meter. 6. TRACK A running track has an inner and outer edge. Both the inner and outer edges consist of two semicircles joined by two straight line segments. The straight line segments are 100 yards long. The radii of the inner edge semicircles are 25 yards each and the radii of the outer edge semicircles are 32 yards each. What is the area of the track? Round your answer to the nearest hundredth of a yard. 7. SEMICIRCLES Bridget arranged three semicircles in the pattern shown. The right triangle has side lengths 6, 8, and 10 inches. a. What is the total area of the three semicircles? Round your answer to the nearest hundredth of a square inch. b. If the right triangle had side lengths 21, 79, and 10 inches, what would the total area of the three semicircles be? Round your answer to the nearest hundredth of a square inch. 16

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19 Learning Targets Students will be able to find lateral areas and surface areas of prisms. Students will be able to find lateral areas and surface areas of cylinders. Section 12.2 Notes: Surface Areas of Prisms and Cylinders Vocabulary, Example Types Definitions, Pictures and Examples Lateral Faces Lateral Edges Altitude Base Edges Height Label the Parts 18

20 Lateral Area Lateral Area of a Prism With a highlighter, Identify the base of the given figures. Example 1: a. b. c. Example 2: The length of each side of the base of a regular octagonal prism is 6 inches, and the height is 11 inches. Find the lateral area. Example 3: Lateral Area of a Prism Find the lateral area of the prism. Round your answers to the nearest hundredth. 19

21 You Try! Example 4: Find the lateral area of the prism. Surface Area of a Prism Surface Area of Prisms Example 5: Find the surface area of the rectangular prism Example 6: Find the surface area of a triangular prism. Round to the nearest tenth. Example 7: Find the surface area of the regular hexagonal prism. 20

22 You Try! Example 8: Find the surface area of the regular pentagonal prism. Lateral Area of a Right Cylinder Surface area of a Cylinder Example 9: Find the lateral area and the surface area of the cylinder. Round to the nearest thousandth. Example 10: Find the lateral area and the surface area of the cylinder. Round to the nearest tenth. 21

23 You Try! Find the lateral area and surface area of each cylinder. Round to the nearest tenth. a. b. Summary! Find the lateral area and surface area of each prism

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25 Geometry Section 12.2 Worksheet Name Find the lateral area and surface area of each prism. Round to the nearest tenth if necessary Lateral Area Lateral Area Surface Area Surface Area Lateral Area Lateral Area Surface Area Surface Area Lateral Area Lateral Area Surface Area Surface Area 24

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27 Geometry Extra Practice 12.2 Name: Surface Areas of Prisms and Cylinders Find the lateral and surface area of each prism. Round to the nearest tenth if necessary Find the lateral area and surface area of each cylinder. Round to the nearest tenth

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29 Learning Targets Students will be able to find lateral areas and surface areas of pyramids. Students will be able to find lateral areas and surface areas of cones. Section 12.3 Notes: Surface Areas of Pyramids and Cones Vocabulary, Example Types Definitions, Pictures and Examples Regular Pyramid Lateral Faces Vertex Lateral Edge Slant Height Label the Regular Pyramid with the given information. 28

30 Lateral Area of a Regular Pyramid Example 1: Find the lateral area of the square pyramid to the nearest tenth. Examples in finding the lateral area of a pyramid. Surface Area of a Regular Pyramid Examples of Surface area of pyramids Example 2: Find the surface area of the square pyramid to the nearest tenth. Example 3: Find the surface area of the regular pyramid. Round to the nearest hundredth. 29

31 You Try! 1. Find the surface area of the square pyramid. 2. Find the surface area of the regular pyramid. Lateral Area of a Cone Example 4: The cone represents a conical slate roof on a house. Find the lateral area. Find the lateral area of a cone Example 5: Find the surface area of the cone. Round to the nearest tenth. Find the surface area of the cone. 30

32 You Try! 3. Find the surface area of the cone. Using the solid and model provided, write in the formulas for each solid. Solid Model Lateral Area Surface Area Prism Cylinder Pyramid Cone 31

33 Geometry 12.3 Worksheet Name: Find the lateral area and surface area of each regular pyramid. Round to the nearest tenth if necessary Find the lateral area and surface area of each cone. Round to the nearest tenth if necessary Find the surface area of a cone if the height is 14 centimeters and the slant height is 16.4 centimeters. 8. Find the surface area of a cone if the height is 12 inches and the diameter is 27 inches. 32

34 9. GAZEBOS The roof of a gazebo is a regular octagonal pyramid. If the base of the pyramid has sides of 0.5 meter and the slant height of the roof is 1.9 meters, find the area of the roof. 10. PAPERWEIGHTS Daphne uses a paperweight shaped like a pyramid with a regular hexagon for a base. The side length of the regular hexagon is 1 inch. The altitude of the pyramid is 2 inches. What is the lateral surface area of this pyramid? Round your answers to the nearest hundredth 33

35 Learning Targets Students will be able to find volumes of prisms. Students will be able to find volumes of cylinders. Section 12.4 Notes: Volumes of Prisms and Cylinders Vocabulary, Example Types Definitions, Pictures and Examples Volume of a Prism Example 1: Find the volume of the prism. Make sure to label the correct units. Example 2: Find the volume of the prism. Make sure you label the correct units. You Try! 1. Find the volume of the prism. Make sure you label the correct units. 34

36 Volume Of a Cylinder Example 2: Find the volume of the cylinder. Make sure to label the correct units and leave your answer in terms of π. You Try! Find the volume of the cylinder. Cavalieri s Principle Example of Cavalieri s Principle Example 3: Find the volume of the oblique cylinder. Label your units and leave your answer in terms of π. 35

37 Summary! Find the volume of the given prism. Find the volume of the cylinder. 36

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39 Geometry 12.4 Worksheet Name: Find the volume of each prism or cylinder. Round to the nearest tenth if necessary. 1. V= 2. V= 3. V= 4. V= 5. V= 6. V= 7. AQUARIUM Mr. Gutierrez purchased a cylindrical aquarium for his office. The aquarium has a height of 25 1 inches and a radius 2 of 21 inches. a. What is the volume of the aquarium in cubic feet? b. If there are 7.48 gallons in a cubic foot, how many gallons of water does the aquarium hold? c. If a cubic foot of water weighs about 62.4 pounds, what is the weight of the water in the aquarium to the "nearest five pounds? 38

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41 Learning Targets Students will be able to find volumes of pyramids. Students will be able to find volumes of cones Section 12.5 Notes: Volumes of Pyramids and Cones Vocabulary, Example Types Definitions, Pictures and Examples Volume of a Pyramid Example 1: Find the volume of the square pyramid. Label the units. Example 2: Find the volume of the triangular pyramid. Label the units. You Try! 1. Find the volume of the pentagonal pyramid. Label the units. Volume of a Cone 40

42 Example 3: Find the volume of the cone. Leave your answers in terms of π and label the units. Example 4: Find the volume of the oblique cone. Round your answers to the tenth. You Try! 1. Find the volume of each cone. In the below chart, fill out the volume formulas for each individual solid. Solid Prism Cylinder Pyramid Cone Model Volume 41

43 Geometry 12.5 Worksheet Name: Find the volume of each pyramid or cone. Round to the nearest tenth if necessary. 1. V= 2. V= 3. V= 4. V= 5. V= 6. V= 7. CONSTRUCTION Mr. Ganty built a conical storage shed. The base of the shed is 4 meters in diameter and the height of the shed is 3.8 meters. What is the volume of the shed? 8. HISTORY The start of the pyramid age began with King Zoser s pyramid, erected in the 27th century B.C. In its original state, it stood 62 meters high with a rectangular base that measured 140 meters by 118 meters. Find the volume of the original pyramid. 9. SCULPTING A sculptor wants to remove stone from a cylindrical block 3 feet high and turn it into a cone. The diameter of the base of the cone and cylinder is 2 feet. What is the volume of the stone that the sculptor must remove? Round your answer to the nearest hundredth. 42

44 10. STAGES A stage has the form of a square pyramid with the top sliced off along a plane parallel to the base. The side length of the top square is 12 feet and the side length of the bottom square is 16 feet. The height of the stage is 3 feet. a. What is the volume of the entire square pyramid that the stage is part of? b. What is the volume of the top of the pyramid that is removed to get the stage? c. What is the volume of the stage? 43

45 Learning Targets Students will be able to find surface areas of spheres. Students will be able to find volumes of spheres. Section 12.6 Notes: Surface Areas and Volumes of Spheres Vocabulary, Example Types Definitions, Pictures and Examples Surface Area of a Sphere Example 1: Find the surface area of the sphere. Round to the nearest tenth and label the units. Find the surface area A plane can intersect a sphere in a point or in a circle. If the circle contains the center of the sphere, the intersection is called a. The endpoints of a diameter of a great circle are called the. Since a great circle has the same center as the sphere and its radii are also radii of the sphere, it is the largest circle that can be drawn on a sphere. A great circle separates a sphere into two congruent halves, called. Surface Area of a Hemisphere Example 2: Find the surface area of the hemisphere. Label the units. 44

46 Example 3: Find the surface area of the sphere if the circumference of the great circle is 10π. Example 4: Find the surface area of the sphere if the area of the great circle is approximately 160 square meters. Volume of a Sphere Example 5: Find the volume of the sphere. Round to the nearest hundredth and label the units. Example 6: Find the volume of the sphere with a great circle that has a circumference of 30π centimeters. Round to the nearest tenth. 45

47 Volume of a Hemisphere Example 7: Find the volume of the hemisphere with a diameter of 6 ft. Example 8 RECESS The jungle gym outside of Jada s school is a perfect hemisphere. It has a volume of 4,000π cubic feet. What is the diameter of the jungle gym? 46

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49 Geometry Name: 12.6 Worksheet Find the surface area of each sphere or hemisphere. Round to the nearest tenth. 1. SA= 2. SA= 3. hemisphere: radius of great circle = 8.4 in. 4. sphere: area of great circle 29.8 m 2 Find the volume of each sphere or hemisphere. Round to the nearest tenth. 5. V= 6. V= 7. hemisphere: diameter = 18 mm 7. V= 8. sphere: circumference 36 yds 8. V= 9. sphere: radius = 12.4 in. 9. V= 2

50 10. CUBES Marcus builds a sphere inside of a cube. The sphere fits snugly inside the cube so that the sphere touches the cube at one point on each side. The side length of the cube is 2 inches. a. What is the surface area of the cube? b. What is the surface area of the sphere? Round your answers to the nearest hundredth. c. What is the ratio of the surface area of the cube to the surface area of the sphere? Round your answer to the nearest hundredth 3

51 Chapter 12 Extension Surface Area of Composite Figures Vocabulary, Example Types Definitions, Pictures and Examples Surface Area of a Composite Figure Identify the solids that form the composite solid Example 1: Identify the solids that make the composite solid and then find the surface area. 4

52 Example 2: Identify the solids that make the composite solid and then find the surface area. Example 3: Steven is making a wood box to fit two flower pots in the middle. His rough draft of the contraption looks like the diagram below. Find the amount of paint he needs to paint the outside and the inside holes. Example 4: Identify the solids that make the composite figure. Find the total surface area and volume of the composite space figures. 5

53 Example 5: Identify the solids that make the composite figure. Find the volume of the composite figure. 1. Find the total surface area and volume of the composite figure. You Try! 2. A storage bin is shaped as in the figure. The radius of the cylindrical top is 7 ft. The overall height of the bin is 26 ft. and the altitude of the conical section is 12 ft. Find the total surface area of the bin in bushels. 6

54 3. The dimensions of the base of a rectangular solid are 5 and 8, and its altitude is 12. A hole, extending from upper base to lower base, is in the shape of a right triangular prism whose bases are equilateral s having an edge of 3. Determine the total surface area of the figure. 4. Jimmy s lunch box in the shape of a half cylinder on a rectangular box. Find the total area of metal needed to manufacture it A rectangular prism is 40 ft by 38 ft by 15 ft. Shown below is the prism with a half cylinder removed. Find the volume of the original prism remaining. 15 ft 38 ft 40 ft 12 ft 10 ft 10 ft 7

CHAPTER 12. Extending Surface Area and Volume

CHAPTER 12. Extending Surface Area and Volume CHAPTER 12 Extending Surface Area and Volume 0 Learning Targets Students will be able to draw isometric views of three-dimensional figures. Students will be able to investigate cross-sections of three-dimensional

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