L22 Measurement in Three Dimensions. 22b Pyramid, Cone, & Sphere

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1 A pyramid (#VOC) is a polyhedron with a polygon base and triangle faces (other than perhaps the base) that meet at the top (apex). There are triangular pyramids, square pyramids, pentagonal pyramids, and more, dependent on the shape of the base. Below is an example of a square pyramid. Label the apex (#VOC) and base shape of this pyramid. Draw the altitude (the shortest line from the apex to the base, which as it turns out is also perpendicular to the base plane). We designate the height of the pyramid to be the length of its altitude (#VOC). Let s investigate the volume of a pyramid. First, let s build a square pyramid. Cut out the outlines of the figure on the following page and fold it (like origami!) to create a pyramid. Note: glue/tape edges are included for your convenience, but are not intended to contribute to the surface area of the pyramid, as they will overlap the faces. We re interested in finding the volume of this pyramid. In groups of four, see if you can arrange enough of these pyramids to form a rectangular prism with equal base and height. How many did you need? What does this tell you about the volume of a single pyramid, in comparison to the volume of the rectangular prism with the same base and height?

2 What is the formula for the volume of the pyramid, in terms of its base measurements and its height? How does this compare to the formula for the volume of the rectangular prism with equal base measurements and height? Although these pyramids are not regular, meaning their sides are not congruent triangles, what argument can we use to justify that the volume of a square pyramid, with fixed base measurements and height, is always the same, regardless of where we locate the apex? Practice Find the volume of a square pyramid the given dimensions. B represents the area of the base of the pyramid. 1. B = 9 inches 2 and h = 7 inches. 2. Note: The fact that the volume of the pyramid is 1/3 the volume of its associated prism is true regardless of the shape of the base (e.g. triangular, 4-sides, 5-sides, etc.)

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4 This page is purposefully left blank to allow for cutouts on the previous page.

5 A cone (#VOC) is a 3-dimensional object with a circular base and one vertex, as shown below to the right. Label the cone s radius (r) and height (h) on the image above. Based on what you know about the relationship between a pyramid and prism with similar dimensions, how do you think the volume of a cone relates to the volume of a cylinder with equal radius and height measurements? The formula for the volume of a cone with radius r and height h is. Why does this seem to make sense based on your square pyramid volume formula? Suppose, for example, that the base of the square pyramid above has the same area as the base of the cone, and each have the same height. That would mean that the cross sections for each, parallel to the base and at equal distances from the base, would also have equal area. What does Cavalieri s principle then tell you, not only about a cone, but about any polygonal pyramid s volume? Similarly, you could imagine a pyramid with an n-gon base, where n was very large, closely resembling a circle. Find the volume of a cone with the given dimensions. 1. r = 4 inches and h = 6 inches Find the volume of a polygonal pyramid with base area B = 10 and h = 12?

6 A sphere (#VOC) is a 3-dimensional object consisting of all points equidistant from a single point, called its center. It has no edges, faces, or vertices. The volume of a sphere with radius r is: V 4 = πr 3 3 Note: You can see this formula works if you have access to a hemisphere with radius r and a cylinder with radius and height both equal to r. The hemisphere will hold 2/3 the amount of water as the cylinder, which means a sphere will hold twice as much, hence the above formula. Practice Find the volume of the sphere with the given dimension. Leave your answers in terms of π or in simplest radical form, AND as a decimal rounded to the nearest thousandths. 1. radius = 8 cm diameter = 6 ft.

7 Homework Name: Per: Date: Find the volume of the sphere with the given dimension. Leave your answers in terms of π or in simplest radical form, AND as a decimal rounded to the nearest thousandths

8 Grandma s birthday is in a few weeks and we would like to buy her a new fish tank for her pet fish, Ramsey and Karyl. Ramsey and Karyl like a lot of water to swim in and therefore Grandma needs a sufficiently large fish tank. There are three tanks available and we want to purchase the one that holds the most water. The first tank is a cylinder with diameter 12 cm and height 30 cm. The second tank is a cone with base diameter 18 cm and height 35 cm. The third tank is a sphere with diameter 20 cm. 1. Which tank should we purchase? 2. How much more water does the largest tank hold than the smallest tank? 3. Suppose the diameter of each tank decreases by 2 cm. Which tank would then hold the most water? 4. The tank company designed a new tank that is shaped like a cylinder on the bottom with a cone on top. The company states that the radius is 12 cm and the total height is 60 cm. a. What would the height of the cylinder portion need to be in order for the total volume to be 7680π cm 3? b. What is the altitude of the cone portion of the tank?