11.4. Imagine that you are, right now, facing a clock and reading the time on that. Spin to Win. Volume of Cones and Pyramids
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1 Spin to Win Volume of Cones and Pyramids.4 Learning Goals In this lesson, you will: Rotate two-dimensional plane figures to generate three-dimensional figures. Give an informal argument for the volume of cones and pyramids. Imagine that you are, right now, facing a clock and reading the time on that clock let s imagine that it s 2:28. Now imagine that you are blasted away from that clock at the speed of light, yet you are still able to read the time on it (of course you wouldn t be able to, really, but this is imagination!). What time would you see on the clock as you traveled away from it at the speed of light? The light bouncing off the clock travels at the speed of light, so, as you travel farther and farther away from the clock, all you could possibly see was the time on the clock as it was right when you left. Einstein considered this thought experiment among many others to help him arrive at groundbreaking theories in physics. 803
2 Problem 1 Building Cylinders You have learned that when two-dimensional shapes are translated, rotated, or stacked, they can form three-dimensional solids. You can also use transformations and stacking to build formulas for three-dimensional figures. 1. Determine the volume for the cylinder shown. Show your work. 8 cm 24 cm To derive the formula for the volume of a cylinder, you can think of the cylinder as an infinite stack of discs, each with an area of pr 2. These discs are stacked to a height of h, the height of the cylinder. So, the volume of the cylinder is pr 2 3 h, or pr 2 h. h r 2 2. Can you choose any disc in the cylinder and multiply the area by the height to calculate the volume of the cylinder? Explain your reasoning. 804 Chapter Three-Dimensional Figures
3 Another way to think about a cylinder is the rotation of a rectangle about one of its sides as shown. The rotation of the set of points that make up the rectangle forms the cylinder. h To determine the volume of the cylinder, you can multiply the area of the rectangle by the distance that the points of the rectangle rotate. However, the points of the rectangle don t all rotate the same distance. Consider a top view of the cylinder. The distance that point A rotates is greater than the distance that point B rotates. r B A You can t calculate the distance that each point rotates because there are an infinite number of points, each rotating a different distance. But you can use an average, or typical point, of the rectangle. 3. Consider the dot plot shown. x x x x x x x x x x x a. What is the median of the data? b. Describe how you could determine the median of these data without doing any calculation. Remember that the median is a measure of center of a data set..4 Volume of Cones and Pyramids 805
4 4. What is the location of the average, or typical, point of the rectangle in terms of the radius and the height? Explain your reasoning. 5. What is the area of the rectangle that is rotated? 6. Use the average point of the rectangle to calculate the average distance that the points of the rectangle rotate. Explain your reasoning. 7. To determine the volume of the cylinder, multiply the area of the rectangle by the average distance that the points of the rectangle rotate. Calculate the volume of the cylinder. 8. Compare the volume that you calculated in Question 2 to the volume that you calculated in Question 7. What do you notice? 806 Chapter Three-Dimensional Figures
5 Problem 2 Building Cones You can also think about a cone as a rotation of a right triangle about one of its legs. You can use the same strategy of determining the average, or typical, point in a right triangle to derive the formula for the volume of a cone. The figure shown represents a cone formed by rotating a right triangle about one of its legs. h Previously, you explored points of concurrency of a triangle. Recall that the centroid is the point of concurrency of the three medians of a triangle. A median connects a vertex to the midpoint of the opposite side. You can use the centroid as the typical point of the triangle to derive the volume. The centroid of the right triangle is shown. r h r.4 Volume of Cones and Pyramids 807
6 Let s review how to calculate the coordinates of the centroid of triangle ABC. Triangle ABC has vertices at A (0, 0), B (0, 18), and C (12, 0). y (0, 18) B D (0, 9) A (6, 0) E (12, 0) x C First, determine the locations of the midpoints of side AB and side AC. Midpoint of side AC: ( , ), or (6, 0) Midpoint of side AB: ( 0 1 0, ), or (0, 9) Next, determine an equation representing each median. The slope of BE is 23, and the y-intercept is 18. Slope of BE So, the equation of BE is y 5 23x The slope of CD is 2 3, and the y-intercept is 9. 4 Slope of CD So, the equation of CD is y x Why do we need to draw only 2 median lines instead of all 3? 808 Chapter Three-Dimensional Figures
7 Solve the system of equations to determine the coordinates of the centroid. y 5 23x x x 1 18 y 5 23x c y x x 5 23x 1 9 y 5 23(4) x 5 x 2 3 y 5 6 x 5 4 The centroid of the triangle is located at (4, 6). y (0, 18) B D (0, 9) (4, 6) A (6, 0) E (12, 0) x C.4 Volume of Cones and Pyramids 809
8 1. What is the area of triangle ABC? 2. What is the average distance that all the points of the triangle are rotated? Show your work and explain your reasoning. 3. Determine the volume of the cone. Show your work and explain your reasoning. 4. Use the formula for the volume of a cone, 1 3 p r 2 h, to calculate the volume of the cone. Show your work. 5. Compare the volume that you calculated by rotating the right triangle to the volume that you calculated using the formula for the volume of a cone. What do you notice? 810 Chapter Three-Dimensional Figures
9 6. Derive the formula for the volume of any cone with radius, r, and height, h, by rotating a right triangle with vertices at (0, 0), (0, h), and (r, 0). y B A C x.4 Volume of Cones and Pyramids 8
10 Problem 3 And Now, Pyramids You can use what you know about the similarities and differences between cylinders and cones to make conjectures about the volume of pyramids. 1. Compare different ways to create cylinders and cones. What similarities and differences are there between creating cylinders and cones: a. by stacking? b. by rotating? 2. Analyze the formulas for the volumes of the cylinder and cone. Volume of cylinder 5 r 2 h Volume of cone r 2 h a. Which part of each formula describes the area of the base of each solid figure, B? Explain why. b. Which part of each formula describes the height of each solid figure? c. Rewrite the volume formulas for each solid figure using the variables B and h. 812 Chapter Three-Dimensional Figures
11 3. How are the formulas for the volumes of a cylinder and cone similar and different? 4. Now compare different ways to create prisms and pyramids. What similarities and differences are there between creating prisms and pyramids: a. by stacking? b. by rotating? Can I create a pyramid or prism by rotating a polygon? 5. Analyze the formula for the volume of a prism. Volume of prism: V 5 Bh a. Which part of the formula represents the area of the base? b. Which part of the formula represents the height? 6. Based on your answers to Questions 1 through 5, what conjecture can you make about the formula for the volume of any pyramid? Explain your reasoning. Be prepared to share your solutions and methods..4 Volume of Cones and Pyramids 813
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