Picture a rectangle divided into two right triangles by a diagonal. How is the area of the right triangle formed by the diagonal related to the area of the rectangle? The area of any rectangle is the product of its width and length. For example, if a rectangle is 3 inches wide and 5 inches long, its area is 15 square inches (length times width). The figure below shows a rectangle "split" along a diagonal, demonstrating that the rectangle can be thought of as two equal right triangles joined together. The areas of rectangles and right triangles are proportional to one another: a rectangle has twice the area of the right triangle formed by its diagonal. In a similar way, the volumes of a cone and a cylinder that have identical bases and heights are proportional. If a cone and a cylinder have bases (shown in color) with equal areas, and both have identical heights, then the volume of the cone is one-third the volume of the cylinder. Imagine turning the cone in the figure upside down, with its point downward. If the cone were hollow with its top open, it could be filled with a liquid just like an ice cream cone. One would have to fill and pour the contents of the cone into the cylinder three times in order to fill up the cylinder. The figure above also illustrates the terms height and radius for a cone and a cylinder. The base of the cone is a circle of radius r. The height of the cone is the length h of the straight line from the cone's tip to the center of its circular base. Both ends of a cylinder are circles, each of radius r. The height of the cylinder is the length h between the centers of the two ends. The volume relationship between these cones and cylinders with equal bases and heights can be expressed mathematically. The volume of an object is the amount of space enclosed within it. For example, the volume of a cube is the area of one side times its height. The figure below shows a cube. The area of its base is indicated in color. Multiplying this (colored) area by the height L of the cube gives its volume. And since each dimension (length, width and height) of a cube is identical, its volume is L × L × L, or L 3, where L is the length of each side. The same procedure can be applied to finding the volume of a cylinder. That is, the area of the base of the cylinder times the height of the cylinder gives its volume. The bases of the cylinder and cone shown previously are circles. The area of a circle is πr 2, where r is the radius of the circle. Therefore, the volume V cyl is given by the equation: V cyl πr 2h (area of its circular base times its height) where r is the radius of the cylinder and h is its height. The volume of the cone (V cone) is one-third that of a cylinder that has the same base and height: . The cones and cylinders shown previously are right circular cones and right circular cylinders, which means that the central axis of each is perpendicular to the base. There are other types of cylinders and cones, and the proportions and equations that have been developed above also apply to these other types of cylinders and cones. Philip Edward Koth with William Arthur Atkins BibliographyAbbott, P. Geometry. New York: David Mckay Co., Inc., 1982. Internet ResourcesThe Method of Archimedes. American Mathematical Society. <http://www.ams.org/new-in-math/cover/archimedes2.html>.
Learning Outcomes
The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone. In geometry, a cone is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See the image below. The height of a cone is the distance between its base and the vertex.  Earlier in this section, we saw that the volume of a cylinder is [latex]V=\text{\pi }{r}^{2}h[/latex]. We can think of a cone as part of a cylinder. The image below shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder. The volume of a cone is less than the volume of a cylinder with the same base and height. In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is Since the base of a cone is a circle, we can substitute the formula of area of a circle, [latex]\text{\pi }{r}^{2}[/latex] , for [latex]B[/latex] to get the formula for volume of a cone. In this book, we will only find the volume of a cone, and not its surface area.
For a cone with radius [latex]r[/latex] and height [latex]h[/latex] .
Find the volume of a cone with height [latex]6[/latex] inches and radius of its base [latex]2[/latex] inches. Solution
Marty’s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is [latex]8[/latex] inches tall and [latex]5[/latex] inches in diameter? Round the answer to the nearest hundredth. In the following video we provide another example of how to find the volume of a cone.
A cone is a three-dimensional figure with one circular base. A curved surface connects the base and the vertex.
The volume of a 3 -dimensional solid is the amount of space it occupies. Volume is measured in cubic units ( in 3 , ft 3 , cm 3 , m 3 , et cetera). Be sure that all of the measurements are in the same unit before computing the volume. The volume V of a cone with radius r is one-third the area of the base B times the height h . V = 1 3 B h or V = 1 3 π r 2 h , where B = π r 2 Note : The formula for the volume of an oblique cone is the same as that of a right one. The volumes of a cone and a cylinder are related in the same way as the volumes of a pyramid and a prism are related. If the heights of a cone and a cylinder are equal, then the volume of the cylinder is three times as much as the volume of a cone.
Example: Find the volume of the cone shown. Round to the nearest tenth of a cubic centimeter.
Solution From the figure, the radius of the cone is 8 cm and the height is 18 cm. The formula for the volume of a cone is, V = 1 3 π r 2 h Substitute 8 for r and 18 for h . V = 1 3 π ( 8 ) 2 ( 18 ) Simplify. V = 1 3 π ( 64 ) ( 18 ) = 384 π ≈ 1206.4 Therefore, the volume of the cone is about 1206.4 cubic centimeters. |
What is the volume of the cone that has the same base and height of the cylinder as shown below?
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