Area and Volume
Area refers to the measure of the surface enclosed by a shape, such as a square or circle, and is typically expressed in square units. Volume, on the other hand, refers to the measure of the space occupied by a three-dimensional object, such as a cube or sphere, and is typically expressed in cubic units. Both area and volume are fundamental concepts in geometry and calculus.
8 Key excerpts on "Area and Volume"
eBook - ePub- Alan J. Cann(Author)
- 2013(Publication Date)
- Wiley(Publisher)
...5 Areas and Volumes LEARNING OBJECTIVES: On completing this chapter, you should be able to: write the formulae for calculating the areas and volumes of two and three-dimensional objects. calculate the areas and volumes of complex shapes. apply this knowledge to biological problems. 5.1. Geometry Geometry is the branch of mathematics that deals with the properties of space and objects. It is one of the oldest branches of mathematics, named from Greek for ‘Earth measurement’. Trigonometry is the branch of mathematics concerned with specific functions of angles and their application to calculations in geometry. Classically, geometry deals with simple, regular shapes (Tables 5.1 and 5.2). 5.2. Calculating areas and volumes Calculations involving simple geometric shapes are usually straightforward (except for ellipsoids) – use the formulae in Tables 5.1 and 5.2. However, it is sometimes said that ‘There are no straight lines in biology’. Most biological shapes are complex, but are either approximations of simple shapes or combinations of simple shapes. Therefore, to determine shapes and volumes in biology, it is necessary to devise a strategy by which to approach the problem. This is a complex area which is impossible to define absolutely in simple terms, but the following examples illustrate some possible approaches. Table 5.1 Equations for perimeter and area of two-dimensional shapes Examples 1. A tissue is composed of cells which are roughly spherical in shape and about 45 μm in diameter. Calculate the total number of cells present in 1 cm 3 of this tissue. Table 5.2 Equations for surface Area and Volume of three-dimensional shapes Calculate the volume of a single cell in μm 3 : Calculate the number of μm 3 in 1 cm 3 of tissue: Calculate the number of cells in 1 cm 3 of tissue: 2. Fibroblast cells like the one illustrated above are cultured on circular glass coverslips 10mm in diameter, and have a doubling time of 24 h...
eBook - ePub- Tandi Clausen-May(Author)
- 2013(Publication Date)
- SAGE Publications Ltd(Publisher)
...Any other triangle can then be seen as a shearing of a right-angled triangle with the same base length, height and area. This ‘model to think with’ can be applied to any parallelogram or triangle, so that the learner can mentally transform the figure on the page back into a rectangle or right-angled triangle. Then the base and height can be identified easily, and so its area may be found. There is no need for a formula: the process is visual and kinaesthetic, not symbolic. PowerPoint PowerPoint 9-2, Areas of Straight-Sided Shapes, shows how to find the area of a parallelogram or a triangle by first shearing it so that it has a right angle, and then basing the calculations on the area of the resulting rectangle. e) Capacity and Volume We have seen that area can be thought of as ‘an amount of flatness’. It is a strictly two-dimensional concept. Capacity and volume, on the other hand, are three-dimensional. But our world is three-dimensional, so these concepts may actually be easier to understand. Capacity may be the best place to start. A capacity relates to a particular container, and it tells you how much that container can hold. This idea may be established using informal measures, such as the number of small cups-full that are needed to fill a big jug. Volume, on the other hand, is the amount of ‘stuff’ that is needed to make a solid. Activities which involve building up cuboids a layer at a time out of cubic-centimetre cubes to find their volumes are commonly used to introduce a more formal measure of volume. This is a good practical approach, and it may be extended in due course to other solids – but it is important to emphasise that the volume relates to the whole block of cubes, not to an empty container. An empty carton has a capacity – but it is empty, so its only ‘volume’ is the volume of plastic or cardboard of which it is composed...
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...Chapter 14 Volumes of common solids Why it is important to understand: Volumes 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 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 Science and Mathematics for Engineering...
eBook - ePub- Christine Hopkins, Ann Pope, Sandy Pepperell(Authors)
- 2013(Publication Date)
- David Fulton Publishers(Publisher)
...The cubes can be seen as building up in layers. Example 1 Example 2 The rectangular face of each layer of the cube or cuboid has rows and columns representing the length and breadth of the rectangle. The number of layers is also the height of the cube or cuboid. Thus: • The volume of a cuboid (or a cube) can be calculated by multiplying length by breadth by height. This is the same as the area of the base multiplied by the height. PRISMS Take a prism with a right-angled triangle as its base. Slide two of these prisms together and you have a cuboid. The volume of the right-angled triangular prism is half that of the cuboid. This idea can be extended to triangular prisms with any type of triangle as base. The area of a triangle was found by drawing the smallest possible rectangle around the triangle with twice the area of the triangle. The volume of a triangular prism can be found by fitting it into a cuboid whose volume is twice that of the triangular prism. The volume of the cuboid is the area of the rectangular cross-section times the length. The volume of the triangular prism is half the area of the rectangular cross-section (that is the area of the triangular face) times the length. It is possible to show that for any prism: • Volume of the prism = area of cross-section × length. MEASURING SURFACE AREA Surface area is the total area of all the surfaces of a solid. So the surface area of a cube of side 2 cm is found by calculating the area of one square face as 2 cm × 2 cm = 4 cm 2. As there are six faces the total surface area is 24 cm 2. Surface area can also be thought of as the area of the net of the solid. MEASURING CAPACITY To measure the volume of some materials, such as a quantity of water, it is necessary to contain them. The volume of the water is then compared to the space within a container, or how much it holds (its capacity)...
eBook - ePubBasic Math Concepts
For Water and Wastewater Plant Operators
- Joanne K. Price(Author)
- 2018(Publication Date)
- Routledge(Publisher)
...11 Volume Measurement SKILLS CHECK Complete and score the following skills test. Each section should be scored separately in the box provided to the right. For Section 11.1, a score of 8 or above indicates you are sufficiently strong in that concept. A score of 7 or below indicates a review of that section is advisable. For Section 11.2, a score of 4 or above indicates you are sufficiently strong in that concept. A score of 3 or below indicates a review of that section is advisable. 11.1 VOLUMES—BASIC SHAPES SUMMARY 1. The general equation for most volume calculations is: Volume = [Representative Surface Area ] [ Third Dimension ] 2. The equations for the four basic volume shapes are as follows: 3. The equations for the volume of a cone and sphere are as follows: A measurement of volume indicates the holding capacity of an object. A simple approach to most volume calculations is to remember that volume is surface area times a third dimension. The surface area used in these calculations must be the representative surface, the side that gives the object its basic shape. In the diagram to the left, the representative surface areas of a rectangle, triangle, trapezoid and circle have been illustrated. These are the four basic shapes most often found in water and wastewater calculations. Calculations of the cone and sphere volume do not use the general volume equation since there is no “representative surface area.” Both of these volume calculations, however, are based on the volume of a cylinder...
eBook - ePubMathematics in the Primary School
A Sense of Progression
- Sandy Pepperell, Christine Hopkins, Sue Gifford, Peter Tallant(Authors)
- 2014(Publication Date)
- Routledge(Publisher)
...The most likely contexts for using these measures will be in other curriculum areas such as science. To measure the volume of solid shapes, a basic unit is required. The most frequently used unit is the cube with sides of 1 centimetre (1 cm 3): note that a volume of 1 cm 3 has a capacity of 1 millilitre; larger volumes can be measured in cubic metres (1 m 3). The move towards an understanding of the volume of cuboids can be made through building models with cubes. Calculating the volume can be developed through systematic counting of the cubes (one layer has four cubes, there are two layers, and so the volume is eight cubes), leading to repeated addition and multiplication. Children could explore the patterns obtained in the numbers of unit cubes used to make different sized cubes. They could be asked to solve a problem of wrapping 12 gift boxes to be sent through the post, using the smallest amount of paper. The different shapes made from the same number of cubes could be explored, to help develop the concept of conservation of volume. If the volume stays the same what does distinguish one shape from another? Area Although this aspect is introduced formally in Key Stage 2, early ideas of area are encountered in covering surfaces; for example, predicting how many sheets of newspaper will be needed to cover the table. Other contexts relevant here are estimating how much wallpaper or carpet is needed in the home corner or in a dolls’ house. The language to be developed will involve the amount of material needed for covering and wrapping. Activities could include wrapping parcels, making tablecloths for the home corner or covers for the Three Bears’ beds. Conservation of area appears to be grasped later than in learning to measure length and so experiences of cutting up and rearranging shapes could be encouraged well before Key Stage 2 (Gifford 2005). In this section there is a strong emphasis on practical contexts...
eBook - ePub- Raymond Paul, Walter Whyte(Authors)
- 2012(Publication Date)
- Routledge(Publisher)
...Chapter 14 Areas and volumes 14.1 Introduction The surveyor is often required to determine areas of land and sometimes to subdivide areas and fix new boundaries. Volume calculations, e.g. in construction earthworks, disused quarries, opencast mining, etc., all involve a knowledge of area calculation methods first. Volume calculations in surveying are generally concerned with the volume of earthworks and may be subdivided into calculations based on cross-sections, used for roads, trenches, etc., and calculations based on contours or spot heights. Computer methods, including the use of digitisers, are not covered here. The standard area units are the square kilometre (km), the hectare (ha) and the square metre (m). All may be used in surveying as appropriate. 10000 m 2 = l ha (approx. 2.5 acres) 100 ha = 1 km 2 = the area of one OS National Grid 1:2500 map sheet 14.2 Areas of simple figures 14.2.1 Rectangle, square Area = length X breadth (14.1) Figure 14.1 14.2.2 Triangle Given base length b and perpendicular height h (Figure 14.1), area = b X h /2 (14.2) Given sides a, b and c and s = (a + b + c)/2, area = √{ s (s – a)(s – b)(s – c)} (14.3) Given two sides, a and b and included angle C, area = (a b sin C)/2 (14.4) Given side a and. angles A, B and C, area = (a 2 sin B sin C)/(2sinA) (14.5) 14.2.3 Trapezium Given parallel sides a and b and perpendicular distance between them A, area = (a + b) h /2 (14.6) 14.2.4 Circle Given radius r and π = 3.141592, area = πr 2 (14.7) 14.2.5 Segment of a circle Figure 14.2 Area of the sector less the area of the triangle in Figure 14.2, area = (π r 2 α/360) - (r 2 sin α)/2 (14.8) Note that (r 2 sin α)/2 comes from Equation (14.4). 14.3 Areas from drawings or plans Areas may be obtained from drawings, from field measurements (with or without plotting) or from calculated data such as rectangular co-ordinates or partial coordinates...
eBook - ePub- Douglas K. Brumbaugh, Peggy L. Moch, MaryE Wilkinson(Authors)
- 2004(Publication Date)
- Routledge(Publisher)
...The third variable, n, tells the number of triangles that are needed, in this case 6. Fig. 5.13 Your Turn 22. Use the formula,, to find the area of a region defined by a regular octagon with sides of length 4 cm and an apothem of length 4.828 cm. Volume Just as area is directly measured using a two-dimensional model, volume can be measured using a three-dimensional model. For example, how many rolls of quarters will it take to exactly fill your sock drawer? You might decide that you need to break your unit roll into 40 subunits or disks in order to completely fill the space and obtain a reasonably close approximation of the volume. Perhaps you would like a more convenient unit, such as a cube. How many sugar cubes will it take to fill your coffee mug? If your mug is a cylinder rather than a right, rectangular-based prism, then you will find that you must develop (or recall) some estimation strategies. As you stuff the cubes in the cup, some of them will be out of sight. In our discussion about area, we talked about covering a region with unit squares. For volume, we could use a unit cube whose face is the same size as the unit square that was used to discuss area. Suppose a rectangle is 8 units long and 3 units wide. From the area work, the rectangle would be covered by exactly 32 unit squares. We could place a unit cube on each of those squares, as shown in Fig. 5.14, and now a wondrous thing has happened. That rectangle we used to discuss area is still visible (the tops of the cubes), but there is now a depth factor as well. Counting the cubes, rather than the top face of each cube, tells us that the figure has a volume of 32 unit cubes. We have a length of 8 units, a width of 4 units, and a height of 1 unit, giving a total of 32 unit cubes. If a second layer of cubes is placed on top of the first, then the length is still 8 units, the width is still 4 units, but the height is now 2 units, and now we have used 64 unit cubes...
