Lesson 4.7 Core Focus on Geometry Volume of Cones

Warm-Up Simplify. 1. 2. 3. 4. 5. A cylinder is 24 cm tall. The area of its base is 17 square centimeters. a. Find the volume of the cylinder. b. What is the volume of a cylinder that is as tall? c. What is the volume of a cylinder that is as tall? 14 8 15 408 cm3 204 cm3 136 cm3

Lesson 4.7 Volume of Cones Find the volume of cones and solve real-world problems involving cones.

Explore! Cones in a Cylinder Rita is making snow cones for her friends. She has four cylindrical glasses that she is using to take the crushed ice outside to her friends that are waiting to fill their snow cone cups. Each snow cone cup is as tall as one of the glasses and has the same size circular base. Rita wants to determine if there is a relationship between a cone and a cylinder. She needs to make sure she has enough ice for 12 snow cones. Complete the Explore! to find if Rita will have enough ice for the snow cones. Step 1 Make or find a cone and a cylinder that have congruent bases and are the same height. Step 2 Estimate how many times larger the volume of the cylinder is compared to the volume of the cone. Step 3 Fill the cone with rice, beans or popcorn kernels. Pour the contents of the cone into the cylinder. Repeat until the cylinder is full. Step 4 How many times did you need to empty the cone in order to fill the cylinder?

Explore! Cones in a Cylinder Step 5 What fraction is the volume of the cone compared to the volume of the cylinder? Step 6 What is the formula for the volume of a cylinder? Step 7 Combine the answers from Step 5 and Step 6 to write a formula that can be used to find the volume of a cone. Step 8 Use your formula to find the volume of each cone. Use 3.14 for . Round to the nearest hundredth. a. b. c. r = 2.1 mm Step 9 Will Rita have enough crushed ice for 12 snow cones if she fills the four cylinders? 6 in 15 in 7.5 ft 18 ft 9.4 mm

Volume of a Cone The volume of a cone is equal to one-third of the product of the area of the base (B) and the height (h). h r

Example 1 4 cm 12 cm 12.6 cm Find the volume of the cone. Use 3.14 for π. Write the volume formula for a cone. Substitute all known values for the variables. Find the value of the power. Multiply. The volume of the cone is about 200.96 cm3.

Example 2 Chantel helped with her sister’s party. Each child received a party hat full of treats. Each hat had a volume of 65.94 cubic inches. The radius of each hat was 3 inches. About how tall was each party hat? Write the volume formula for a cone. Substitute all known values for the variables. Find the value of the power & multiply. Divide both sides of the equation by 9.42. Each party hat was about 7 inches tall. 9.42 9.42

Only the positive root is used in this situation. Example 3 A cone-shaped popcorn container holds 314 cubic inches of popped corn. The container is 12 inches tall. Find the radius of the conical container. Use 3.14 for . Write the volume formula for a cone. Substitute all known values for the variables. Multiply then divide both sides by 12.56. Square root both sides of the equation. The radius of the popcorn container is about 5 inches. 12.56 Only the positive root is used in this situation.

Communication Prompt Cleo has 4 congruent cylinders. Cleo also has one cone that is the same height and has a base congruent to the base of the cylinders. She needs to use the cone to fill up the cylinders. Use a picture to show how Cleo could fill up the cylinders using the cone.

Exit Problems 1. Find the volume of the cone. Use 3.14 for . 2. A cone has a volume of 1,899.7 cubic inches. The height of the cone is 15 inches. Find the length of the radius. Use 3.14 for π. 6 m 8 m 10 m ≈ 301.44 cubic meters 11 inches