Unit 7: Area and Volume
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1 Unit 7: Area and Volume Name Math 8, Period Rectangular Prism Triangular Prism Cylinder Cone Sphere Concepts and Skills to be mastered: By the end of this section students should be able to: 1. Find the volume of a cylinder, cone, and sphere given a radius and height. 2. Find a missing measurement (height, radius, or diameter) for a cylinder, cone, or sphere given the volume. 3. Use the formulas for the volumes of cylinders, cones, and spheres to solve a variety of real-world problems. For this unit round all answers to the nearest hundredths.
2 FORMULAS: AREA: Square Rectangle Triangle Parallelogram Circle VOLUME: Rectangular Prism Triangular Prism Cylinder Cone Sphere Unit 7: Volume pg. 1
3 Perimeter and Area: Perimeter is Area is Find the perimeter and area of each shape Unit 7: Volume pg. 2
4 Find the length of the missing side then find the perimeter Unit 7: Volume pg. 3
5 4. Find area and perimeter. 5. Find area and perimeter. 6. Find area and perimeter. 7. Find the area of the shaded. Unit 7: Volume pg. 4
6 PRACTICE: Find the volume of each rectangular prism. a. b. c. d. e. f. Unit 7: Volume pg. 5
7 Volume of Triangular Prisms: PRACTICE: Unit 7: Volume pg. 6
8 Volume of Cylinders: Directions: Find the volume for each cylinder described below. If needed draw and label a picture in. 3 in. 2.5 yd 7 yd 3. Cylinder with a Radius = 21 mm and Height = 19 mm. 4. Cylinder with a Diameter = 8.8 cm and Height = 9 cm. Directions: Find the missing measurement for each cylinder described below. 5. The volume of a cylinder is cubic feet, 6. The volume of a cylinder is 4,224.8 cubic and its height is 15 ft. Find the diameter of millimeters, it has a diameter of 16.4 mm, the base of the cylinder. find the height of the cylinder. Extension: Find the circumference of the base of the cylinder. Unit 7: Volume pg. 7
9 Directions: For each problem given below draw and label a picture that describes each cylinder. Then solve the problem. 7. An ice cream company wants to package a pint of ice cream in a circular cylinder that is 4 inches high. A pint is 16 fluid ounces and 1 fluid ounce is 1.8 cubic inches. What does the radius of the base circle have to be? 8. For a science project, Hassan put a can out to collect rainwater. The can was 11 inches tall and had a diameter of 8 inches. If it rained exactly 20 cubic inches each day, how many days did it take to fill the can? Unit 7: Volume pg. 8
10 Homework: Volume of Cylinders Directions: Find the volume for each cylinder described below. If needed draw and label a picture cm 14 mm 40 mm 2.5 cm 3. Cylinder with a radius of 2 ft and a height of 7 ft. 4. Cylinder with a diameter of 2.7 m and a height of 30 m. Find the missing measurement for each cylinder described below. 5. The volume of a cylinder is 63.6 cubic inches, 6. The volume of a cylinder is 8,685.9 cubic ft, it and its height is 9 inches. Find the diameter of has a diameter of 19.2 ft, find the height of the the base of the cylinder. cylinder. Extension: Find the circumference of the base of the cylinder. Directions: For each problem given below draw and label a picture that describes each cylinder. Then solve the problem. 1. What is the volume of Keisha s thermos if it has a radius of 2.5 in at the opening and 10 in for a height? 2. Mr. Riley bought 2 cans of paint to paint his garage. Each can had a radius of 5.5 inches and a height of 8 inches. How many cubic inches of paint did he buy in all? Unit 7: Volume pg. 9
11 Volume of Cones: Directions: Find the volume for each cone described below. If needed draw and label a picture cm 8 ft The volume of the cone is 5 ft 15 cm The volume of the cone is 3. A cone with a radius of 8.4 feet and a height of 5.5 feet. 4. A cone with a diameter of 9 meters and a height of 4.2 meters. Directions: Find the missing measurement for each cone described below. Round your answer to the nearest tenth. 5. The volume of a cone is cubic inches, and 6. The volume of a cone is cubic ft, it has a its height is 4.5 inches. Find the diameter of the diameter of 12 ft, find the height of the cylinder. base of the cone. For each problem given below draw and label and picture that describes each cone. Then solve the problem. 7. Salt and sand mixtures are often used on icy roads. When the mixture is dumped from a truck into the staging area, it forms a cone-shaped mound with a diameter of 10 feet and a height of 6 feet. What is the volume of the salt-sand mixture? Unit 7: Volume pg. 10
12 8. A glass in the shape of a cone has a diameter of 8 cm. If the glass has a volume of 200 ml (or 200 cubic centimeters), what is the greatest depth that a liquid can be poured into the glass? Explain. Homework: Volume of Cones Directions: Find the volume for each cone described below. If needed draw and label a picture yd 20 cm 2 yd 45 cm 3. A cone with a radius of 40 feet and a height of 100 feet. 4. A cone with a diameter of 4.2 meters and a height of 5 meters. Directions: Find the missing measurement for each cone described below. 5. The volume of a cone is 37.7 cubic inches, and its 6. The volume of a cone is cubic ft, it has a height is 4 inches. Find the diameter of the base diameter of 20 ft, find the height of the cone. of the cone. Unit 7: Volume pg. 11
13 Directions: For each problem given below draw and label a picture that describes each cylinder. Then solve the problem. 7. The American Heritage Center at the University of Wyoming is a conical building. If the height is 77 feet, and the area of the base is about 38,000 square feet, find the volume of air that the heating and cooling systems would have to accommodate. 8. A stalactite, a geological formation, in the Endless Caverns in Virginia is cone-shaped. It is 4 feet long and has a diameter at its base of 1.5 feet. a. Assuming that the stalactite forms a perfect cone, find the volume of the stalactite. b. The stalactite is made of calcium carbonate, which weighs 131 pounds per cubic foot. What is the weight of the stalactite? Unit 7: Volume pg. 12
14 Volume of Spheres: Directions: Find the volume for each sphere described below. If needed draw and label a picture A sphere with a radius of 1.3 yds. 4. A sphere with a diameter of 25 inches Directions: Find the missing measurement for each sphere described below. Round your answer to the nearest tenth. 5. The volume of a sphere is in 3 ; find the diameter of the sphere. 6. The volume of a sphere is ft 3 ; find the radius of the sphere. Directions: For each problem given below draw and label a picture that describes each sphere. Then solve the problem. 7. If a golf ball has a diameter of 4.3 centimeters and a tennis ball has a diameter of 6.9 centimeters, find the difference between the volumes of the two balls. Unit 7: Volume pg. 13
15 8. Kauri pours the water out of a cylindrical flower vase with a height of 5 inches and a radius of 4 inches into a spherical flower vase. The spherical vase has a radius of 4 inches. Will the water overflow? If so, by how much? If not, how much space is left in the spherical vase? 8.3d Homework: Volume of Spheres Directions: Find the volume for each sphere described below. If needed draw and label a picture A sphere with a radius of 10 yards. 4. A sphere with a diameter of 60 inches Directions: Find the missing measurement for each sphere described below. 5. The volume of a sphere is cm 3 ; find the 6. The volume of a sphere is 4,188.8 cubic feet; find diameter of the sphere. the radius of the sphere. Directions: For each problem given below draw and label a picture that describes each sphere. Then solve the problem. 7. The diameter of the moon is 3,476 kilometers. Approximate the volume of the moon. 8. Find the volume of the empty space in a cylindrical tube of three tennis balls. The diameter of each ball is 2.5 inches. The cylinder is 2.5 inches in diameter and is 7.5 inches tall. Unit 7: Volume pg. 14
16 Class Activity: Volume of Cylinders, Cones, and Spheres Task 1: Silos A silo is a storage bin that is a cylinder with a hemisphere on top. A farmer has a silo with a base radius of 30 feet and a storage height of 100 feet. The storage height is the part which can be filled with grain - it is just the cylinder. A cubic foot of grain weighs 62 lbs. a. Draw and label a picture of the silo b. How many pounds of grain can the farmer store in the silo? c. How high (including the hemispherical top) is the silo? d. One thousand square feet of wheat produces 250 pounds of grain. The farmer s wheat field is 3,500 feet by 20,000 feet. Is the silo large enough to hold the grain? By how much? Explain your answer. e. If the farmer decides to fill the silo all the way to the top of the hemisphere how many cubic feet of grain can he store? Unit 7: Volume pg. 15
17 Task 2: Snow Cones A snow cone consists of a cone filled with flavored shaved ice topped with hemisphere of flavored shaved ice. The cone is 4 inches long and the top has a diameter of 3 inches. a. Draw and label a picture of the snowcone. b. How much shaved ice, in cubic inches, is there altogether? c. If 6 cubic inches of flavored ice is equal to 1 ounce, how many ounces of shaved ice is that? d. If one ounce of flavored shaved ice is 50 calories, how many calories will you consume if you eat this snow cone? Unit 7: Volume pg. 16
18 Task 3: Pipes Which will carry the most water? Explain your answer. Two pipes each 100 cm tall. One with a 3 cm radius and the other with a 4 cm radius One pipe that is also 100 cm tall with a 5 cm radius. Task 4: Fruit A cantaloupe a diameter of 23 centimeters and a Clementine orange has a diameter of 7 centimeters. Predict how many times bigger the cantaloupe is than the orange. Then calculate the volume of each fruit to determine how many times bigger the cantaloupe is than the orange. Unit 7: Volume pg. 17
19 Homework: Volume of Cylinders, Cones, and Spheres Task 1: Containers A cylindrical glass 7 cm in diameter and 10 cm tall is filled with water to a height of 9 cm. If a ball 5 cm in diameter is dropped into the class and sinks to the bottom, will the water in the glass overflow? If it does overflow, how much water will be lost? Explain and justify your response. Task 2: Ice Cream Izzi s Ice Cream Shoppe is about to advertise giant spherical scoops of ice cream 8 cm in diameter! Izzi wants to be sure there is enough ice cream and wonders how many scoops can be obtained from each cylindrical container of ice cream. The containers are 20 cm in diameter and 26 cm tall. a. Draw and label a picture of the ice cream containers and the scoop of ice cream. b. Determine the number of scoops of ice cream one container will give her? c. Ingrid purchases one of these famous giant scoops of ice cream but does not get to it fast enough and the ice cream melts! The radius of the cone and the ice cream (sphere) is 4 cm and the height of the cone is 10 cm. Will all of the melted ice cream fit inside the cone? d. If it does fit, how much more ice cream will fit in the cone? If it doesn t fit, how many cubic centimeters of ice cream does she need to eat before it melts in order to make it fit? Unit 7: Volume pg. 18
20 1. 2. Unit 7: Volume pg. 19
21 Find the volume. Unit 7: Volume pg. 20
22 6. What is the volume of the empty space left when three tennis balls with a diameter of 3 inches are packed into a cylindrical container that is 9 inches tall with the same diameter as the tennis ball? 7. Unit 7: Volume pg. 21
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