ESSENTIAL QUESTION How do you find the volume of a cone? 8.6.B
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1 ? LESSON 9.2 Volume of Cones ESSENTIAL QUESTION How do you find the volume of a cone? Expressions, equations, and relationships 8.6.B Model the relationship between the volume of a cylinder and a cone... and connect that relationship to the formulas. 8.7.A Solve problems involving the volume of cones. EXPLORE ACTIVITY 8.6.B Modeling the Volume of a Cone A cone is a three-dimensional figure that has one vertex and one circular base. To explore the volume of a cone, Sandi does an experiment with a cone and a cylinder that have congruent bases and heights. She fills the cone with popcorn kernels and then pours the kernels into the cylinder. She repeats this until the cylinder is full. Sandi finds that it takes 3 cones to fill the volume of the cylinder. STEP 1 STEP 2 STEP 3 What is the formula for the volume V of a cylinder with base area B and height h? What is the area of the base of the cone? Sandi found that, when the bases and height are the same, times V cone = V cylinder. STEP 4 How does the volume of the cone compare to the volume of the cylinder? Volume of the cone: V cone = V cylinder Reflect 1. Use the conclusion from this experiment to write a formula for the volume of a cone in terms of the height and the radius. Explain. 2. How are the formulas for the volume of a cone and a pyramid similar? Lesson
2 Math On the Spot Finding the Volume of a Cone Using a Formula The formulas for the volume of a prism and the volume of a cylinder are the same: multiply the height h by the area of the base B, so V = Bh. In the Explore Activity, you saw that the volume of a cone is one third the volume of a cylinder with the same base and height. Volume of a Cone The volume V of a cone with radius r is one third the area of the base B times the height h. h V = 1 3 Bh or V = 1 3 π r 2 h r My Notes EXAMPLE A Find the volume of each cone. Round your answers to the nearest tenth. Use 3.14 for π. A V = 1_ 3 π r 2 h 8 in. 1_ _ Substitute. Simplify. Multiply. B 2 in. The volume is about 33.5 in 3. Since the diameter is 8 ft, the radius is 4 ft. V = 1_ 3 π r 2 h 9 ft 1_ _ Substitute. Simplify Multiply. 8 ft The volume is about ft Unit 3 Reflect 3. How can you rewrite the formula for the volume of a cone using the diameter d instead of the radius r?
3 YOUR TURN Find the volume of each cone. Round your answers to the nearest tenth. Use 3.14 for π cm 5. 3 ft 2 ft 16 cm Personal Math Trainer Online Assessment and Intervention Finding the Volume of a Volcano The mountain created by a volcano is often cone shaped. EXAMPLE 2 For her geography project, Karen built a clay model of a volcano in the shape of a cone. Her model has a diameter of 12 inches and a height of 8 inches. Find the volume of clay in her model to the nearest tenth. Use 3.14 for π. 8.7.A Math On the Spot Image Credits: Marco Regalia Sell/Alamy Images STEP 1 STEP 2 Find the radius. r = 12 2 = 6 in. Find the volume of clay. V = 1_ 3 π r 2 h 1_ _ Substitute. Simplify. Multiply. The volume of the clay is about in 3. YOUR TURN 6. The cone of the volcano Parícutin in Mexico had a height of 410 meters and a diameter of 424 meters. Approximate the volume of the cone. Personal Math Trainer Online Assessment and Intervention Lesson
4 Guided Practice 1. The area of the base of a cylinder is 45 square inches and its height is 10 inches. A cone has the same area for its base and the same height. What is the volume of the cone? (Explore Activity) 2. A cone and a cylinder have congruent height and bases. The volume of the cone is 18 m 3. What is the volume of the cylinder? Explain. (Explore Activity) Vcylinder = Bh = = Vcone = 1 3 V cylinder = 1 3 = The volume of the cone is in 3. Find the volume of each cone. Round your answer to the nearest tenth if necessary. Use 3.14 for π. (Example 1) ft 100 in. 6 ft 33 in.? 5. Gretchen made a paper cone to hold a gift for a friend. The paper cone was 15 inches high and had a radius of 3 inches. Find the volume of the paper cone to the nearest tenth. Use 3.14 for π. (Example 2) 6. A cone-shaped building is commonly used to store sand. What would be the volume of a cone-shaped building with a diameter of 50 meters and a height of 20 meters? Round your answer to the nearest tenth. Use 3.14 for π. (Example 2) ESSENTIAL QUESTION CHECK-IN 7. How do you find the volume of a cone? For help, use the model in the Explore Activity. 254 Unit 3
5 Name Class Date 9.2 Independent Practice Find the volume of each cone. Round your answers to the nearest tenth if necessary. Use 3.14 for π B, 8.7.A 7 mm Personal Math Trainer Online Assessment and Intervention 13. A snack bar sells popcorn in cone-shaped containers. One container has a diameter of 8 inches and a height of 10 inches. How many cubic inches of popcorn does the container hold? 8 mm 14. A volcanic cone has a diameter of 300 meters and a height of 150 meters. What is the volume of the cone? 9. 6 in. 15. Multistep Orange traffic cones come in a variety of sizes. Approximate the volume, in cubic inches, of a traffic cone that has a height of 2 feet and a diameter of 10 inches. Use 3.14 for π. 2 in. Find the missing measure for each cone. Round your answers to the nearest tenth if necessary. Use 3.14 for π. 10. A cone has a diameter of 6 centimeters and a height of 11.5 centimeters. 11. A cone has a radius of 3 meters and a height of 10 meters. Round your answers to the nearest tenth if necessary. Use 3.14 for π. 12. Antonio is making mini waffle cones. Each waffle cone is 3 inches high and has a radius of 3 _ 4 inch. What is the volume of a waffle cone? 16. radius = height = 6 in. volume = in diameter = 6 cm height = volume = cm The diameter of a cone-shaped container is 4 inches, and its height is 6 inches. How much greater is the volume of a cylindershaped container with the same diameter and height? Round your answer to the nearest hundredth. Use 3.14 for π. Lesson
6 FOCUS ON HIGHER ORDER THINKING Work Area 19. Alex wants to know the volume of sand in an hourglass. When all the sand is in the bottom, he stands a ruler up beside the hourglass and estimates the height of the cone of sand. a. What else does he need to measure to find the volume of sand? b. Make a Conjecture If the volume of sand is increasing at a constant rate, is the height increasing at a constant rate? Explain. 20. Problem Solving The diameter of a cone is x cm, the height is 18 cm, and the volume is cm 3. What is x? Use 3.14 for π. 21. Analyze Relationships A cone has a radius of 1 foot and a height of 2 feet. How many cones of liquid would it take to fill a cylinder with a diameter of 2 feet and a height of 2 feet? Explain. 22. Critique Reasoning Herb knows that the volume of a cone is one third that of a cylinder with the same base and height. He reasons that a cone with the same height as a given cylinder but 3 times the radius should therefore have the same volume as the cylinder, since 1_ 3 = 1. Is Herb 3 correct? Explain. 256 Unit 3
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Write Euler s Theorem. Solving Problems Using Surface Area and Volume. Figure Surface Area Volume. Cl V 5 1 } 3
CHAPTER SUMMARY Big Idea 1 BIG IDEAS Exploring Solids and Their Properties For Your Notebook Euler s Theorem is useful when finding the number of faces, edges, or vertices on a polyhedron, especially when
Study Guide and Review
State whether each sentence is or false. If false, replace the underlined term to make a sentence. 1. Euclidean geometry deals with a system of points, great circles (lines), and spheres (planes). false,
Geometry SOL G.13 G.14 Area, Surface Area, Volume Study Guide
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For full credit, show all work. Label all answers. For all problems involving a formula, show the formula and each step to receive full credit.
Accelerated Review 1: Pythagorean Theorem/Dim Geo II Name: For full credit, show all work. Label all answers. For all problems involving a formula, show the formula and each step to receive full credit.
Determine whether the solid is a polyhedron. If it is, name the polyhedron. Explain your reasoning
Chapter 12 Review Packet Name Determine whether the solid is a polyhedron. If it is, name the polyhedron. Explain your reasoning. 1. 2. 3. Use Euler's Theorem to find the value of n. Faces: 10 Vertices:
C in. 2. D in Find the volume of a 7-inch tall drinking glass with a 4-inch diameter. C lateral faces. A in. 3 B in.
Standardized Test A For use after Chapter Multiple Choice. Which figure is a polyhedron? A B 7. Find the surface area of the regular pyramid. A 300 ft 2 B 340 ft 2 C 400 ft 2 C D D 700 ft 2 2. A polyhedron
12-3 Surface Areas of Pyramids and Cones
18. MOUNTAINS A conical mountain has a radius of 1.6 kilometers and a height of 0.5 kilometer. What is the lateral area of the mountain? The radius of the conical mountain is 1.6 kilometers and the height
Determine the surface area of the following square-based pyramid. Determine the volume of the following triangular prism. ) + 9.
MPM 1D Name: Unit: Measurement Date: Calculating and of Three Dimensional Figures Use the Formula Sheet attached to help you to answer each of the following questions. Three problems are worked out for
8th Grade. 3-Dimensional Solids. Slide 1 / 97 Slide 2 / 97. Slide 3 / 97. Slide 3 (Answer) / 97. Slide 4 / 97. Slide 5 / 97.
Slide / 97 Slide 2 / 97 8th Grade D Geometry 205--20 www.njctl.org Slide / 97 Table of ontents Slide () / 97 Table of ontents -Dimensional Solids Volume Prisms and ylinders Pyramids, ones & Spheres More
Skills Practice Skills Practice for Lesson 6.1
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Unit 3 Surface Area and Volume
Name: 3.1 Areas of 2D Figures 3.2 Surface Areas of Prisms and Pyramids 3.3 Surface Areas of Cylinders, Cones and Spheres 3.4 Volumes of Prisms and Pyramids 3.5 Volumes of Cylinders, Cones and Spheres 3.1
Converse of the. Pythagorean Theorem 8.7.C. Decide whether the converse of the Pythagorean Theorem is true.
LESSON 8. Converse of the Pythagorean Theorem ESSENTIAL QUESTION Expressions, equations, and relationships Use the Pythagorean Theorem and its converse to solve problems. How can you test the converse