Mathematics
Volume of Solid
The volume of a solid is the measure of the amount of space it occupies. In mathematics, it is typically calculated using formulas specific to the shape of the solid, such as the volume of a cube, cylinder, or sphere. Understanding volume is important for various applications, including geometry, physics, and engineering.
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6 Key excerpts on "Volume of Solid"
eBook - PDF- Richard Aufmann, Joanne Lockwood, Richard Nation, Daniel K. Clegg(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
We are now ready to introduce volume of geometric solids. Geometric solids are three-dimensional shapes that are bounded by surfaces. Com- mon geometric solids include the rectangular solid, sphere, cylinder, cone, and pyramid. Despite being called “solids,” these figures are actually hollow; they do not include the points inside their surfaces. Volume is a measure of the amount of space occupied by a geometric solid. Volume can be used to describe, for example, the amount of trash in a landfill, the amount of con- crete poured for the foundation of a house, or the amount of water in a town’s reservoir. A rectangular solid is one in which all six sides, called faces, are rectangles. The variable L is used to represent the length of a rectangular solid, W is used to represent its width, and H is used to represent its height. A shoe- box is an example of a rectangular solid. L H W Color: Brown Size: 7M Comfort Shoes Comfort Shoes Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SECTION 7.5 | Volume and Surface Area 407 A cube is a special type of rectangular solid. Each of the six faces of a cube is a square. The variable s is used to represent the length of one side of a cube. A baby’s block is an example of a cube. A cube that is 1 ft on each side has a volume of 1 cubic foot, which is written 1 ft 3 . A cube that measures 1 cm on each side has a volume of 1 cubic centimeter, writ- ten 1 cm 3 . The volume of a solid is the number of cubes, each of volume 1 cubic unit, that are necessary to exactly fill the solid.
eBook - PDF- John Bird(Author)
- 2017(Publication Date)
- Routledge(Publisher)
Chapter 20 Volumes and surface areas of common solids Why it is important to understand: Volumes and surface areas of common solids There are many practical applications where volumes and surface areas of common solids are required. Examples include determining capacities of oil, water, petrol and fish tanks, ventilation shafts and cooling towers, determining volumes of blocks of metal, ball-bearings, boilers and buoys, and calculating the cubic metres of concrete needed for a path. Finding the surface areas of loudspeaker diaphragms and lampshades provide further practical examples. Understanding these calculations is essential for the many practical applications in engineering, construction, architecture and science. At the end of this chapter, you should be able to: • state the SI unit of volume • calculate the volumes and surface areas of cuboids, cylinders, prisms, pyramids, cones and spheres • calculate volumes and surface areas of frusta of pyramids and cones • calculate the frustum and zone of a sphere • calculate volumes of regular solids using the prismoidal rule • appreciate that volumes of similar bodies are proportional to the cubes of the corresponding linear dimensions 20.1 Introduction The volume of any solid is a measure of the space occupied by the solid. Volume is measured in cubic units such as mm 3 , cm 3 and m 3 . This chapter deals with finding volumes of common solids; in engineering it is often important to be able to calculate volume or capacity to estimate, say, the amount of liquid, such as water, oil or petrol, in differing shaped containers. A prism is a solid with a constant cross-section and with two ends parallel. The shape of the end is used to describe the prism. For example, there are rectangular prisms (called cuboids), triangular prisms and circular prisms (called cylinders).
eBook - ePub- John Bird(Author)
- 2021(Publication Date)
- Routledge(Publisher)
Chapter 28Volumes and surface areas of common solids
Why it is important to understand: Volumes and surface areas of common solidsThere are many practical applications where volumes and surface areas of common solids are required. Examples include determining capacities of oil, water, petrol and fish tanks, ventilation shafts and cooling towers, determining volumes of blocks of metal, ball-bearings, boilers and buoys, and calculating the cubic metres of concrete needed for a path. Finding the surface areas of loudspeaker diaphragms and lampshades provide further practical examples. Understanding these calculations is essential for the many practical applications in engineering, construction, architecture and science.At the end of this chapter you should be able to:- state the SI unit of volume
- calculate the volumes and surface areas of cuboids, cylinders, prisms, pyramids, cones and spheres
- calculate volumes and surface areas of frusta of pyramids and cones
- appreciate that volumes of similar bodies are proportional to the cubes of the corresponding linear dimensions
28.1 Introduction
The volume of any solid is a measure of the space occupied by the solid. Volume is measured in cubic units such asandmm 3,cm 3m 3.This chapter deals with finding volumes of common solids; in engineering it is often important to be able to calculate volume or capacity to estimate, say, the amount of liquid, such as water, oil or petrol, in different shaped containers.A prism is a solid with a constant cross-section and with two ends parallel. The shape of the end is used to describe the prism. For example, there are rectangular prisms (called cuboids), triangular prisms and circular prisms (called cylinders).On completing this chapter you will be able to calculate the volumes and surface areas of rectangular and other prisms, cylinders, pyramids, cones and spheres, together with frusta of pyramids and cones. Volumes of similar shapes are also considered.
- Richard Lehrer, Leona Schauble(Authors)
- 2023(Publication Date)
- Routledge(Publisher)
5 EXTENDING MOTION TO THREE DIMENSIONS Volume and Its MeasureDOI: 10.4324/9781003287476-5In typical elementary school mathematics curricula, learning about volume measurement is initiated by acquiring a formula for calculating the measure of right rectangular prisms: volume = length × width × height. Often, this approach fails to engage students’ conceptual understanding of volume because it requires only that students recall and execute a simple arithmetic procedure (Simon & Blume, 1994 ). Evidence suggests that approaches that emphasize volume calculation are less successful than those that foster a conceptual understanding of solid volume measurement (Huang & Wu, 2019 ). A conceptual grasp of volume is important so that students can extend volume measure beyond the traditional focus on rectangular prisms to include prisms with other bases and cylinders and other settings of volume measure where the familiar formula no longer applies. Moreover, as in other realms of spatial measure, it is critical to help students develop a theory of measure organized around core concepts and anchored in practical activity.Structuring and Dynamic Approaches to Volume Measure
Accordingly, mathematics education researchers have explored two general approaches to fostering students’ intuitions and emerging conceptions of volume. The first, extensively pursued by Battista, Clements, and colleagues (Barrett, Clements, & Sarama, 2017 ; Battista, 1990 , 1999 , 2004 , 2007 , 2012 ; Battista & Clements, 1996 , 1998 ; Clements, Swaminathan, Hannibal, & Sarama, 1999 ; O’Dell et al., 2017 ), supports students’ structuring of rectangular prisms and compositions of prisms as a lattice of unit cubes. This entails first enacting and eventually imagining completely filling 3-D structures with cubic units of a standard size and enumerating the cubes to obtain a measure of volume. Some (e.g., Panorkou, 2020 ) refer to this form of imagining as a packing or filling model, to reflect its emphasis on identifying the amount of substance that an object can hold. Battista and Clements (1996)
- Frank R. Spellman, Nancy E. Whiting(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
Sphere— A container shaped like a ball. Square units— Measurements used to express area (e.g., square feet, square meters, acres). Volume— The capacity of a unit (how much it will hold), measured in cubic units (e.g., cubic feet, cubic meters) or in liquid volume units (e.g., gallons, liters, million gallons). Width— The distance from one side of the tank to the other, measured in linear units. 61 Basic Math Operations 2.11.2 R ELEVANT G EOMETRIC E QUATIONS Circumference C of a circle: C = π d = 2 π r Perimeter P of a square with side a : P = 4 a Perimeter P of a rectangle with sides a and b : P = 2 a + 2 b Perimeter P of a triangle with sides a , b , and c : P = a + b + c Area A of a circle with radius r ( d = 2 r ): A = π d 2 /4 = π r 2 Area A of duct in square feet when d is in inches: A = 0.005454 d 2 Area A of a triangle with base b and height h : A = 0.5 bh Area A of a square with sides a : A = a 2 Area A of a rectangle with sides a and b : A = ab Area A of an ellipse with major axis a and minor axis b : A = π ab Area A of a trapezoid with parallel sides a and b and height h : A = 0.5( a + b ) h Area A of a duct in square feet when d is in inches: A = π d 2 /576 = 0.005454 d 2 Volume V of a sphere with a radius r ( d = 2 r ): V = 1.33 π r 3 = 0.1667 π d 3 Volume V of a cube with sides a : V = a 3 Volume V of a rectangular solid (sides a and b and height c ): V = abc Volume V of a cylinder with a radius r and height H : V = π r 2 h = π d 2 h /4 Volume V of a pyramid: V = 0.33 2.11.3 G EOMETRICAL C ALCULATIONS 2.11.3.1 Perimeter and Circumference On occasion, it may be necessary to determine the distance around grounds or landscapes. To mea-sure the distance around property, buildings, and basin-like structures, it is necessary to determine either perimeter or circumference. The perimeter is the distance around an object; a border or outer boundary. Circumference is the distance around a circle or circular object, such as a clarifier.
eBook - PDFSpellman's Standard Handbook for Wastewater Operators
Volume I, Fundamental Level, Second Edition
- Frank R. Spellman(Author)
- 2010(Publication Date)
- CRC Press(Publisher)
Proper design and operational control requires the engineer and operator to perform several process control calculations. Many of these calculations include parameters such as the circumference or perim-eter, area, or volume of the tank or channel as part of the information necessary to determine the result. Many process calculations require computation of surface areas. To aid in performing these calculations, the following definitions and relevant equations used to calculate areas and volumes for several geometric shapes are provided. 3.10.1 Definitions Area —The area of an object, measured in square units. Base —The term used to identity the bottom leg of a triangle, mea-sured in linear units. Circumference —The distance around an object, measured in linear units. When determined for other than circles, it may be called the perimeter of the figure, object, or landscape. Cubic units —Measurements used to express volume, cubic feet, cubic meters, etc. Depth —The vertical distance from the bottom of the tank to the top. It is normally measured in terms of liquid depth and given in terms of sidewall depth (SWD), measured in linear units. Diameter —The distance, measured in linear units, from one edge of a circle to the opposite edge passing through the center. Height —The vertical distance, measured in linear units, from one end of an object to the other. Length —The distance, measured in linear units, from one end of an object to the other. Linear units —Measurements used to express distance (e.g., feet, inches, meters, yards). Pi (π)—A number used in calculations involving circles, spheres, or cones (π = 3.14). Radius— The distance, measured in linear units, from the center of a circle to the edge. 3 Basic Mathematics 45 Sphere— A container shaped like a ball. Square units— Measurements used to express area (e.g., square feet, square meters, acres).
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