VOLUME of CONES Essential Question? How do you find the volume of a cone? 8.G.9

Common Core Standard: 8.G ─ Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres. 9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Objectives: To find the volume of a cone.

Volume of Cones A CONE is a three- dimensional figure that has one vertex and one circular base. RECALL: The VOLUME of any three-dimensional figure is the number of cubic units needed to fill the space taken up by the solid figure.

Volume of Cones Think of a cone inside a cylinder with the same base and height. What do you know about the volume of the cone in comparison to the cylinder? The volume of the cone inside the cylinder is LESS THAN the volume of the cylinder What kind of number would we multiply the volume of the cone by to get something less than the original?

EXPERIMENT How many cones did it take to fill the cylinder? From the geometric solid kit, take the large cone and one cylinder. Use the top (plastic circle) to verify that the interior base of the cylinder and the interior base of the cone are the same size. Place both the cone and the cylinder on the desk and place a book on top to verify that they are the same height. Make a prediction as to how many cones would fit in to the cylinder. Now CAREFULLY fill the cone with water. Pour the water from the cone into the cylinder. CAREFULLY repeat Steps 5 and 6 until the cylinder is full. How many cones did it take to fill the cylinder? Write a sentence describing the relationship of the volume of the cylinder in relationship to the number of cones it took to fill the cylinder.

What was the sentence you wrote. You need 3 cones to fill 1 cylinder What was the sentence you wrote? You need 3 cones to fill 1 cylinder. How would you write this as an equation? Volume of 3 cones = Volume of 1 cylinder Now solve to find the volume of a cone. V (3 cones) = V (1 cylinder) V 3 cones 3 = V 1 cylinder 3 V (1 cone) = 1 3 V (1 cylinder) What is the formula for the volume of a cylinder? 𝑉 cylinder =𝜋 𝑟 2 ℎ Use substitution to write the formula for the volume of a cone: 𝑉 cone = 1 3 𝜋 𝑟 2 ℎ

We have discovered that the formula for the Volume of Cones We have discovered that the formula for the VOLUME OF A CONE is 𝑽 𝒄𝒐𝒏𝒆 = 𝟏 𝟑 𝑩𝒉 or 𝑽 𝒄𝒐𝒏𝒆 = 𝟏 𝟑 𝝅 𝒓 𝟐 𝒉

Using the Volume of a Cone Formula Find the volume of each cone. Round your answers to the nearest tenth if necessary. Use 3.14 for π.

Using the Volume of a Cone Formula Find the volume of each cone. Round your answers to the nearest tenth if necessary. Use 3.14 for π.

Using the Volume of a Cone Formula For her science 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.

Using the Volume of a Cone Formula V3Solar has developed a cone-shaped solar energy harvester that is claimed to generate over 20 times more electricity than a flat panel. The radius of the base of each harvester is 3 meters and the height of each harvester is 10 meters. How much cubic space does each harvester occupy?