Lesson 4: Volumes of Pyramids and Cones
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1 : Volumes of Pyramids and Cones Learning Targets I can calculate the volumes of pyramids. I can apply the properties of right triangles and trigonometry to find the volume of pyramids Volumes of pyramids Definition: Volume is the space that a three-dimensional figure can occupy. Ex: The volume of a glass is the amount of liquid it can hold. The volume of a pyramid Example 1. Find the volume of the pyramid V= (1/3) Area Base * height or V = The volume of the cone Example 2. Find the volume of the cone V =
2 Applications of Right Triangles and Trigonometry To calculate the volume of the right cone we might be given slant height, height and the radius and the angle formed between the slant height and radius Example 3. Find the volume of the cone when the slant height is 12 and radius is 4 Example 4. What is the volume of the cone? Example 5 What if the volume of the cone that is part of this water tower If FD = 8.5 ft and, determine and state, to the nearest cubic foot, the volume of the water tower.
3 : Volumes of Pyramids and Cones Classwork 1. Find the volume of each pyramid a. b. 2. Find the volume of the cone in terms of. a. b. c. 3. The cone and the cylinder have the same base and the same height. If the cone is filled with liquid and poured into the cylinder repeatedly, how many times would it take to fill the cylinder (explain your answer)
4 : Volumes of Pyramids and Cones Day 2 Composite Figures 1. Find the volume of the composite figure. 2. Many of the ancient pyramids of Egypt are still buried in the sand. Satellites can now locate some of these buried structures using advanced technology. One of the buried pyramids has a height if 250 feet and square base with a side length of 340 feet. How much sand, to the nearest thousand cubic feet, is displaced by the pyramid? 3. A rectangular pyramid fits exactly on top of a rectangular prism. The prism has a length of 12 cm, a width of 6 cm, and a height of 8 cm. The pyramid has a height of 10 cm. Find the volume of the composite figure. 4. A conical grain storage tank has a height of 30 meters and a diameter of 14 meters. Find the capacity of the storage tank. Round the answer to the nearest square meter.
5 5. The vertex of a right cone has a 20 o angle and the slant height is 10 cm. Find the volume of the cone. 6. Jessica and Michele are working on a science lab project where they are creating salt solutions. Jessica s container is in the shape of a rectangular prism which is 2 inches wide, 4 inches long and 8 inches high. She fills it with water up to the 6 inch mark and adds 15 grams of salt to the water. Michele s container is right triangular prism whose triangular base has legs that are 3 and 6 inches, respectively. The height of her container is also 8 inches. She fills in with water to the 7 inch mark and adds 15 grams of salt to the water. Which container has the greatest concentration of salt? Justify your solution 7. Suppose you fill a conical paper cup with a height of 6" with water. If all the water is then poured into a cylindrical cup with the same radius and same height as the conical paper cup, to what height will the water reach in the cylindrical cup? 8. Sand falls from a conveyor belt and forms a pile on a flat surface. The diameter of the pile is approximately 10 ft. and the height is approximately 6 ft. Estimate the volume of the pile of sand. State your assumptions used in modeling.
6 : Volumes of Pyramids and Cones Day 2 Day 2 - Problem Set 1. Identify the solid shown, and find its volume. 2. Find the volume of the right rectangular pyramid shown. 3. Find the volume of the circular cone in the diagram. (Use 22 as an approximation of pi.) 7
7 4. Find the volume of the cone. 5. If a cone has height 9 ft and volume 75 ft 3, find the radius of the base. 6. The vertex of a right cone has a 40 o angle and the slant height is 12 cm. Find the volume of the cone.
3. Draw the orthographic projection (front, right, and top) for the following solid. Also, state how many cubic units the volume is.
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