Mathematics
Surface Area of a Solid
The surface area of a solid is the total area of all the surfaces of the solid. It is calculated by finding the sum of the areas of all the individual surfaces. In mathematics, surface area is an important concept used to measure the amount of material needed to cover the outside of a three-dimensional object.
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3 Key excerpts on "Surface Area of a 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).
- Jean-Pierre Ollivier, Jean-Michel Toorenti, Myriam Carcasses(Authors)
- 2012(Publication Date)
- Wiley-ISTE(Publisher)
Chapter 3Specific Surface Area of Materials3.1. Definition
The specific surface area (or volume) of a material is the ratio expressed in m−1 between the outer surface S of the material and its volume. It can also be expressed in relation to its mass, m . This is known as the specific surface mass or specific surface area. It is this quantity, known asThe outer surface is the material surface in contact with the external environment, whether it is liquid or gas.Ss, that we will consider later and that we will denote, as per current practice, by the term specific surface area .It is usually expressed in m2 /kg or in cm2 /g.The specific surface area may be defined for granular materials and also for porous materials, which are studied in the second part of this book. For a granular material, the smaller the grains, the greater the ratio of outer surface to unit mass and so the larger the specific surface area. This quantity may therefore be used to account for the fineness of grains.The value of the specific surface area of a material generally depends on the method of measurement used, which must be specified. For example, we will thus speak of the Blaine or BET specific surface area, etc.3.1.1. The importance of this parameter: Portland cement hydration
Portland cement is in the form of a powder. By mixing it with water, we obtain a fluid cement paste. Cement grains react with water to form hydrate compounds, which structure themselves over time. The cement paste stiffens then the solid formed progressively hardens. Cement hydration may be described by the degree of hydration, which, in a given time, is equal to the percentage of hydrated cement. Figure 3.1 shows the progression of this degree of hydration over time in relation to the fineness of the cement.Figure 3.1.Influence of the specific surface area of cement on the degree of hydration. The black dots correspond to cement whose grains have an average size of 5 μm and the gray dots to cement whose average size is 30 μm (according to [BEN 99])
eBook - PDF- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
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 • appreciate that volumes of similar bodies are proportional to the cubes of the corresponding linear dimensions 14.1 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 differently 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. Volumes of similar shapes are also considered. 14.2 Calculating volumes and surface areas of common solids 14.2.1 Cuboid or rectangular prism A cuboid is a solid figure bounded by six rectangular faces; all angles are right angles and opposite faces are Science and Mathematics for Engineering. 978-0-367-20475-4, © John Bird. Published by Taylor & Francis. All rights reserved. 144 Section I h b l Figure 14.1 equal. A typical cuboid is shown in Figure 14.1 with length l, breadth b and height h. Volume of cuboid = l × b × h and surface area = 2 bh + 2 hl + 2 lb = 2 (bh + hl + lb) A cube is a square prism.
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