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Exploring the Volume and Surface Area of Cones: Key Concepts and Examples

In this lesson, we will explore the volume and surface area of cones through engaging examples and practice problems. Learn how to sketch and label a cone, calculate its surface area, and determine its volume using the formula ( V = frac{1}{3} pi r^2 h ). We will work through real-life applications, including calculating the height of a cone-shaped party hat from its volume. Finally, we will compare cones and cylinders to understand their relationships in geometry. Dive into the world of geometry with these practical exercises!

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Exploring the Volume and Surface Area of Cones: Key Concepts and Examples

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  1. Lesson 21 Surface Area & Volume Volume of Cones

  2. Warm-Up • Sketch and label a diagram of a cone with a radius of 3 cm and a height of 7 cm. • Find the surface area of the cone in #1. • A cone and a cylinder have congruent bases and are the same height. Predict what the relationship between their volumes will be.

  3. Volume of Cones Target: Calculate the volume of cones.

  4. 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).

  5. Example 1 Find the volume of the cone. Use 3.14 for π. • Write the formula. V = πr2h • Substitute known values. V ≈ (3.14)(4)2(12) • Find the value of the power. V ≈ (3.14)(16)(12) • Multiply. V ≈ 200.96 • The volume of the cone is about 200.96 cm3.

  6. 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. How tall was each party hat? • Write the formula. V = πr2h • Substitute known values. 65.94 ≈ (3.14)(3)2h • Find the value of the power. 65.94 ≈ (3.14)(9)h • Multiply. 65.94 ≈ 9.42h • Divide by 9.42 on both sides. 9.42 9.42 7 ≈ h • Each party hat was about 7 inches tall.

  7. Exit Problems • Find the volume of the cone. Use 3.14 for . • A cone has a volume of 30 cubic inches. What is the volume of a cylinder with the same height and a congruent base to the cone?

  8. Communication Prompt What is similar about cones and pyramids? What is different?

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