Understanding Volumes of Different Shapes
Understanding the volumes of different shapes involves calculating how much space is enclosed within various solid figures. This concept is essential in geometry, where formulas can provide efficient means to determine volume for shapes such as right rectangular prisms, cylinders, pyramids, cones, and spheres. Each of these shapes has a unique formula derived from their dimensions, such as length, width, height, and radius.
Mathematical techniques, including the method of exhaustion and Cavalieri's principle, are employed to derive these volume formulas. The method of exhaustion fills a solid with increasingly smaller shapes to approximate its volume, while Cavalieri's principle compares the volumes of solids based on their cross-sectional areas.
For example, a right circular cylinder can be understood as a prism with a circular base, while a right circular cone is a pyramid with a circular base. The volume of a sphere is determined using Cavalieri’s principle, comparing it to a cylinder and a cone. Overall, mastering the calculation of volumes can enhance one's understanding of geometry and its applications in real-world scenarios.
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Understanding Volumes of Different Shapes
Volume is the amount of space inside a solid figure. For many solid figures the volume can be expressed as a simple formula. For example, a right rectangular prism can be expressed with length l, width w, and height h (Figure 1).
The volume V of the prism is:
The objective of this article is to explore volume formulas for other solids in terms of (1), namely cylinders, pyramids, cones, and spheres. This will be done in part using two mathematical methods of antiquity: the method of exhaustion and Cavalieri’s principle.
The method of exhaustion is used to find the volume of a figure by filling it with figures of ever-decreasing size whose volumes are known. Cavalieri's principle states that if two solid regions are between two parallel planes, and if every plane parallel to the two planes intersects both regions in cross sections of equal area, then the two solids have equal volume. These methods are both early precursors to modern integral calculus.
Overview
A right circular cylinder is obtained by extruding a circular base of radius r to a height h perpendicularly to the base (Figure 2).
This is essentially a prism with a circular base. Referring back to the volume of Figure 1 and rewriting (1) as
, the product in parentheses is the area of the base of the prism. The area of any prism is
, where A is the area of the base. For Figure 2
, so its volume is:
A right rectangular pyramid is obtained by joining each vertex of a rectangular base with the apex, a point directly above the center of the base (Figure 3).
The volume of Figure 3 is obtained by the method of exhaustion. Consider a step pyramid made of n concentric, stacked prisms (Figure 4).
As the number n of prisms increases, the volume of Figure 4 converges to the volume of Figure 3. The kth prism has height
, length
, and width
. The volume of the kth prism is
. So the volume of Figure 4 is:
Letting
in (3), the volume of Figure 3 is obtained:
A right circular cone is essentially a pyramid with a circular base (Figure 5).
The volume of Figure 5 can also be obtained by the method of exhaustion using a stack of concenctric cylinders (Figure 6).
Cylinder k has height
, radius
. By (2), the volume cylinder k is
. The volume of Figure 6 is then:
Letting
, the volume of Figure 5 is obtained.
A sphere is a solid in which every point on the boundary is the same distance r (radius), from a fixed point (center). See Figure 7.
The volume of Figure 7 is obtained by Cavalieri's principle. See Figure 8.
The cone and cylinder both have height and radius r, and the hemisphere has radius r. A plane cuts a cross section parallel to their bases at an altitude y. The cross-sectional area inside the cylinder but outside the cone is
, which equals the cross-sectional area in the hemisphere. So, the volume of the hemisphere equals the volume inside the cylinder but outside the cone, which by (2) and (5) is
.
The entire sphere has volume
, or:
Bibliography
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Clark, David M. Euclidean Geometry: A Guided Inquiry Approach. Berkeley, CA: American Mathematical Society, 2012.
Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee Stiff. Geometry. Evanston, IL: Holt, 2011.
Larson, Ron, and Bruce H. Edwards. Calculus. Boston: Cengage, 2014.
Ostermann, Alexander, and Gerhard Wanner. Geometry by Its History. New York: Springer, 2012.