Mathematics
Volume of prisms
The volume of a prism is the measure of space enclosed by the prism. It is calculated by multiplying the prism's base area by its height. For a rectangular prism, the volume formula is V = lwh, where l is the length, w is the width, and h is the height. This concept is fundamental in geometry and is used to solve real-world problems involving three-dimensional shapes.
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11 Key excerpts on "Volume of prisms"
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
In this triangular prism, the base is the triangle and the height is the distance between the triangular bases (25 cm). 7 cm 10 cm 25 cm 10 cm 10 cm Copyright 2016 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. Unless otherwise noted, all content on this page is © Cengage Learning Section 9.3 Surface Area and Volume 545 Volume = (Area of Base)(Height) Since we have a triangular base, we use the formula Area of Base = 1 2 (base)(height). Note here that we are using the words “base” and “height” to mean different things. We have the Base and Height of the prism, and the base and height of the triangle. Using capital letters when we are talking about the three-dimensional figure (the prism) and lowercase letters when we are talking about the two-dimensional figures (the triangle) can help us differentiate them. Volume 5 (Area of Base)(Height) 5 ( 1 2 bh )(Height) 5 ( 1 2 3 10 3 10)(25) 5 1250 cubic centimeters UNDERSTANDING THE VOLUMES OF PYRAMIDS AND CONES We will illustrate the formula of a pyramid using a cube. It so happens that three congruent pyramids will fit inside the cube. Most people need to see this to believe it, and Figure 9.24 provides the necessary information to make three congruent pyramids that can be joined together to make a cube. Two-dimensional drawings of three-dimensional objects can be difficult to visualize. Geosolids are manipulatives that enable you to demonstrate concepts related to volume. You can fill a geosolid pyramid with water and see that it takes three of them to fill up a geosolid cube with the same height and same size base.
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
A cube is a right square prism whose edges are congruent. SSG EXS. 3–7 DEFINITION As we shall see, the cube is very important in determining the volume of a solid. VOLUME OF A PRISM To introduce the notion of volume, we recognize that a prism encloses a portion of space. Without a formal definition, we say that the volume of the solid is a number that measures the amount of enclosed space. To begin, we need a unit for measuring volume. Just as the meter can be used to measure length and the square yard can be used to measure area, a cubic unit is used to measure the amount of space enclosed within a bounded region of space. One such unit is described in the following paragraph. The volume enclosed by the cube shown in Figure 9.9 is 1 cubic inch or 1 in 3 . The volume of a solid is the number of cubic units within the solid. Thus, we assume that the volume of any solid is a positive number of cubic units. x x 8 cm Figure 9.8 2 bases 4 lateral faces (dividing by 2) (factoring) (reject 221 as a solution) 1 in. 1 in. 1 in. Figure 9.9 9.1 ■ Prisms, Area, and Volume 405 The simplest space figure for which we can determine volume is the right rectangular prism. Such a solid might be described as a parallelpiped or as a “box.” Because boxes are used as containers for storage and shipping (such as a boxcar), it is important to calcu- late volume as a measure of capacity. A right rectangular prism is shown in Figure 9.10; its dimensions are length /, width w, and height h. The volume of a right rectangular prism of length 4 in., width 3 in., and height 2 in. is easily shown to be 24 in 3 . The volume is the product of the three dimensions of the given solid. We see not only that 4 # 3 # 2 5 24, but also that the units of volume are in. # in. # in. 5 in 3 . Figure 9.11(a) and (b) illustrate that the 4 by 3 by 2 box must have the volume 24 in 3 . We see that there are four layers of blocks, each of which is a 2 by 3 configuration of 6 in 3 .
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
He has two circles of the same size and wants to know if he can make both cones from these circles or if one circle needs to be larger than the other. How would you respond? VOLUME Prisms The volume of a three-dimensional figure is a measure of the amount of space that it occupies. To determine the volume of a three-dimensional prism, we imagine the fig- ure filled with unit cubes. A rectangular prism whose sides measure 2, 3, and 4 units, respectively, can be filled with 2 3 4 24 ¥ ¥ = unit cubes (Figure 13.61). The volume of a cube that is 1 unit on each edge is 1 cubic unit. Hence the volume of the rectangular prism in Figure 13.61 is 24 cubic units. As with units of area, we can subdivide 1 cubic unit into smaller cubes to deter- mine volumes of rectangular prisms with dimensions that are terminating decimals. For example, we can subdivide our unit of length into 10 parts and make a tiny cube whose sides are 1 10 of a unit on each side (Figure 13.62). It would take 10 10 10 1000 ¥ ¥ = of these tiny cubes to fill our unit cube. Hence the volume of our tiny cube is 0.001 cubic unit. This subdivision procedure can be used to motivate the following volume formula, which holds for any right rectangular prism whose sides have real number lengths. 4 2 3 Figure 13.61 0 1 Figure 13.62 Section 13.4 Volume 697 From the formula for the volume of a right rectangular prism, we can immediately determine the volume of a cube, since every cube is a special right rectangular prism with all edges the same length. Volume is reported in cubic units. Reflection from Research Learning how to visualize how many cubes are contained in a rectangular box can be complex for students. Having students make predictions of box contents based on pictures of boxes may be critical in helping them see these objects more abstractly (Battista, 1999). Volume of a Right Rectangular Prism The volume V of a right rectangular prism whose dimensions are positive real numbers a b , , and c is V abc = .
- Kevin Corner, Leslie Jackson, William Embleton(Authors)
- 2013(Publication Date)
- Thomas Reed(Publisher)
In other words, it is the ratio of the density of the substance to the density of pure water. Prisms A regular prism is a bar of regular cross-section, some examples are given in Figure 10.1. 236 • Mathematics Figure 10.1 In all these cases, Volume = Area of cross-section × Length Hence, to find the volume of a prism, calculate the area of the end and multiply this by length (or height) of the prism. Example A brass bar 250 mm long has a constant hexagonal cross-section measuring 90 mm across the face from one corner to the opposite corner. Find (i) the volume of the bar, (ii) the mass in kg if the density of brass is 8.4 g/cm 3 . 60° 45 mm 45 mm 45 mm 90 mm Figure 10.2 Side of each equilateral triangle = 45 mm (see Figure 10.2) Perpendicular height = 45 × sin 60 ◦ Area of each equilateral triangle = 1 2 × Base × Perpendicular height Volume – Mass, Centre of Gravity, Moment • 237 = 1 2 × 45 × 45 sin 60 ◦ = 876. 85 mm 2 Therefore area of end face = 6 × 876.85 = 5261.1 mm 2 Volume of prism = Area × Length Therefore V = 5261. 1 × 250 = 1315275 mm 3 = 1315 cm 3 (nearest whole number) Mass ( m ) = Volume ( V ) × Density ( p ) Therefore m = 1315 × 8.4 = 11046 g So mass = 11.05 kg to 2 decimal places. Pyramids A pyramid is a body standing on a triangular, square or polygonal base, its sides tapering to a point at the apex, some examples are illustrated in Figure 10.3. The cone may be considered as a pyramid with a circular base. Figure 10.3 The volume of a pyramid is one-third of the volume of its circumscribing prism. 238 • Mathematics Therefore, the volume of a cone is one-third of the volume of a solid cylinder having the same height and base diameter as the cone. Therefore, volume of a cone = π · r 2 · h 3 . Similarly, the volume of a square pyramid is one-third the volume of a bar of square base with height equal to that of the pyramid.
eBook - PDF- John Bird(Author)
- 2017(Publication Date)
- Routledge(Publisher)
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 rectan- gular and other prisms, cylinders, pyramids, cones and spheres, together with frusta of pyramids and cones. Also, volumes of similar shapes are considered. Engineering Mathematics. 978-1-138-67359-5, © 2017 John Bird. Published by Taylor & Francis. All rights reserved. Volumes and surface areas of common solids 183 20.2 Volumes and surface areas of regular solids A summary of volumes and surface areas of regular solids is shown in Table 20.1. Table 20.1 (i) Rectangular prism (or cuboid) h b l Volume = l × b × h Surface area = 2(bh + hl + lb) (ii) Cylinder h r Volume = πr 2 h Total surface area = 2πrh + 2πr 2 (iii) Pyramid h Volume = 1 3 × A × h where A = area of base and h = perpendicular height Total surface area = (sum of areas of triangles forming sides) + (area of base) (iv) Cone Volume = 1 3 πr 2 h Curved surface area = πrl Total surface area = πrl + πr 2 h r l (v) Sphere r Volume = 4 3 πr 3 Surface area = 4πr 2 20.3 Worked problems on volumes and surface areas of regular solids Problem 1. A water tank is the shape of a rectangular prism having length 2 m, breadth 75 cm and height 50 cm. Determine the capacity of the tank in (a) m 3 (b) cm 3 (c) litres Volume of rectangular prism = l × b × h (see Table 20.1) (a) Volume of tank = 2 × 0.75 × 0.5 = 0.75 m 3 (b) 1m 3 = 10 6 cm 3 , hence 0.75 m 3 = 0.75 × 10 6 cm 3 = 750 000 cm 3 (c) 1 litre = 1000 cm 3 , hence 750 000 cm 3 = 750 000 1000 litres = 750 litres Problem 2.
- Richard Lehrer, Leona Schauble(Authors)
- 2023(Publication Date)
- Routledge(Publisher)
Shortly after these initial qualitative explorations, students begin to find and compare volumes by employing units. As with length and surface area, an important conceptual achievement in understanding volume is coming to understand the properties of units of volume. Students recognize that the units used to measure must be the same or, if not, distinctly labeled, and units need to tile, or completely fill, the volume being measured. Students’ work at this level begins with rectangular prisms. Later, when they encounter other 3-D shapes (such as cylinders), the nature of units may need to be reconsidered, because students have a tendency to favor units that bear a perceptual resemblance to the figure being measured. For example, they may be drawn to using beads to fill the volume of a cylinder because both the beads and the cylinder are “curvy.” However, students usually recall readily that they have already resolved this issue of resemblance in the context of area and quickly reject the solution based on similarity because it does not meet the criteria of avoiding both “gaps” and “overlaps” of measure so that units entirely fill the three-dimensional space.As they first attempt to measure the volume of rectangular prisms, students find volume by counting unit cubes in structures in which all the cubes are visible. Much of the instruction in the early grades is aimed toward helping students differentiate the area of 3-D figures from their volume, but also to perceive the relations between these two attributes in any three-dimensional shape. In Volume unit 1 of Measuring and Visualizing Space, students compare the surface areas of three different “apartment buildings” (that is, prisms) constructed of interlocking cubes. Each building is composed of 12 cubic “apartments” with the following dimensions: 1 × 1 × 12 cubes, 2 × 2 × 3 cubes, and 6 × 1 × 2 cubes. Surface area is contextualized as the windows, roof, and footprint of the building. Students next compare the volume enclosed or occupied by each building—that is, the number of “apartments” that each building holds. In this introductory task, the context supports children in differentiating surface area (e.g., the number of windows, roof, and building footprint) from volume (e.g., the total number of apartments enclosed in each building). Students also find that even though the surface areas of the three buildings vary, their volume measures are the same. The lesson was originally designed for classes of second-grade students, but it can also serve as an entrée to volume measure in later grades.
No longer available |Learn more- Daniel C. Alexander, Geralyn M. Koeberlein, , , Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
3 A cube is a right square prism whose edges are congruent. DEFINITION 1 in. 1 in. 1 in. Figure 9.9 Copyright 2013 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. The simplest space figure for which we can determine volume is the right rectangular prism . Such a solid might be described as a parallelpiped or as a “box.” Because boxes are used as containers for storage and shipping (such as a boxcar), it is important to calculate volume as a measure of capacity. A right rectangular prism is shown in Figure 9.10; its dimensions are length , width w , and height h . The volume of a right rectangular prism of length 4 in., width 3 in., and height 2 in. is easily shown to be 24 in . The volume is the product of the three dimensions of the given solid. We see not only that but also that the units of volume are . Figures 9.11(a) and (b) illustrate that the 4 by 3 by 2 box must have the volume 24 in . We see that there are four layers of blocks, each of which is a 2 by 3 configuration of 6 in . Figure 9.11 provides the insight that leads us to our following postulate. Figure 9.11 (a) (b) 3 3 in. in. in. in 3 4 3 2 24 3 9.1 ■ Prisms, Area, and Volume 395 Unless otherwise noted, all content on this page is © Cengage Learning. Corresponding to every solid is a unique positive number V known as the volume of that solid. POSTULATE 24 ■ Volume Postulate w h Figure 9.10 The frozen solids found in ice cube trays usually approximate the shapes of cubes.
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.
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
Chapter 14 Volumes of common solidsWhy it is important to understand: Volumes 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
- 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 mm3 , cm3 and m3 .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
Science and Mathematics for Engineering. 978-0-367-20475-4, © John Bird. Published by Taylor & Francis. All rights reserved.14.2.1 Cuboid or rectangular prism
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
In practi-cal work, perhaps the most widely used solid is the prism. A prism is a polyhedron that has two identical (congruent) parallel polygon faces called bases and parallel lateral edges. The other sides or faces of a prism are parallelograms called lateral faces . A lateral UNIT 62 VOLUMES OF PRISMS AND CYLINDERS 465 edge is the line segment where two lateral faces meet. An altitude of the prism is a per-pendicular segment that joins the planes of the two bases. The height of the prism is the length of an altitude. Prisms are named according to the shape of their bases, such as triangular, rect-angular, pentagonal, hexagonal, and octagonal. Some common prisms are shown in Figure 62-2. The parts of the prisms are identified. In a right prism , the lateral edges are perpendicular to the bases. Prisms A and B are examples of right prisms. In an oblique prism , the lateral edges are not perpendicular to the bases. Prisms C and D are examples of oblique prisms. 8.20 cm 6.40 cm 15.60 cm FIGURE 62-3 FIGURE 62-2 A. Right Rectangular Prism Lateral Face Base Base Lateral Edge ~ Altitude Base Altitude B. Right Hexagonal Prism Lateral Face Base Lateral Edge ~ C. Oblique Rectangular Prism Lateral Face Base Base Lateral Edge ~ Altitude D. Oblique Triangular Prism Lateral Face Base Base Lat eral Edge ~ Altitude VOLUMES OF PRISMS The volume of any prism (right or oblique) is equal to the product of the base area and height. V 5 A B h where V 5 volume A B 5 area of base h 5 height Example 1 Compute the volume of a prism that has a base area of 34.40 square inches and a height of 16.00 inches. V 5 34.40 sq in. 3 16.00 in. 5 550.4 cu in. Ans Example 2 A solid steel wedge is shown in Figure 62-3. a. Find the volume of the wedge. Round the answer to the nearest tenth cubic centimeter. b. The steel used for the wedge weighs 0.0080 kilogram per cubic centimeter. Find the weight of the wedge. Round the answer to the nearest tenth kilogram.
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
The three faces now lie on a flat surface or plane as shown in Figure 28–24. It can be seen that the total area of the three prism faces is equal to the area of the rectangle in Figure 28–24. Distance A 1 B 1 C is the perimeter of the triangular prism base. Distance A 1 B 1 C is also the length of the rectangle formed by unfolding the prism faces. LENGTH EQUAL TO PERIMETER OF PRISM BASE EDGE F EDGE E EDGE G EDGE F EDGE G ALTITUDE A B C A C B LEFT FACE RIGHT FACE LEFT FACE RIGHT FACE Figure 28–23 Figure 28–24 The lateral area of a right circular cylinder is equal to the product of the cir-cumference of the base and height. LA 5 C B h where LA 5 lateral area C B 5 circumference of base h 5 height Copyright 2019 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. 712 Section V GEOMETRIC FIGURES: AREAS AND VOLUMES This formula is developed in much the same way as that of the prism. Imagine that a vertical cut is made along the lateral surface of the cylinder in Figure 28–25. The lateral surface is then unrolled and spread out flat as shown in Figure 28–26. The length of the rectangle in Figure 28–26 is equal to the circumference of the cylinder base in Figure 28–25. ALTITUDE r LENGTH EQUAL TO CIRCUMFERENCE OF CYLINDER BASE Figure 28–25 Figure 28–26 Surface Areas The surface area of a prism or a cylinder must include the area of both bases as well as the lateral area. The surface area of a prism or a cylinder equals the sum of the lateral area and the two base areas.
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