Volume of Pyramid

4 Key excerpts on "Volume of Pyramid"

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  eBook - ePub

  The Everything Everyday Math Book

  From Tipping to Taxes, All the Real-World, Everyday Math Skills You Need

  • Christopher Monahan(Author)
  • 2013(Publication Date)
  • Everything

    (Publisher)

  Area measurements are given in square units. When you multiply feet by feet, the answer is in square feet.

  Volume

  There are essentially three shapes that make up the basic solids whose volume formulas are known: the prism, the pyramid, and the sphere. The prism is a solid whose top and bottom (the bases) are exactly the same shape and size. The pyramid can have any shape as the base, but all the lateral edges come to a point (similar to the pyramids in Egypt—the difference between the pyramids in Egypt and the mathematical definition of a pyramid is that in math the base does not have to be a rectangle). The sphere is a ball (or a three-dimensional circle, if you prefer).

  Three special cases of solids are usually identified. A prism with all sides of equal length, and all angles right angles, is called a cube . A prism with a circular base is called a cylinder . A pyramid with a circular base is called a cone . Because the base for the prism and the pyramid can be any shape, the formula for the volume is given in terms of the area of the base, B .

  Volume of a Cube

  A = s 3

  Volume of a Prism

  A = B × h

  Volume of a Pyramid

  A = × B × h

  Volume of a Cylinder

  A = π × r 2 × h

  Volume of a Sphere

  A = × π × r 3

  Volume of a Cone

  A = × π × r 2 × h

  Whereas area is measured in square units, volume is measured in cubic units.

  Example: Find the volume of a cube that measures 5 cm on each side.

  The volume of a cube is s 3 , so the volume of this cube is 5 × 5 × 5 = 125 cubic centimeters, or 125 cu. cm, or 125 cm3 . (In the medical field, the cubic centimeter is often abbreviated as cc.)

  Example: The base of a rectangular box has dimensions 8 in. by 12 in. What is the height of the box if the volume of the box is 480 in3 ?

  The base of the box is a rectangle, and the area of a rectangle is l × w . For this box, the area of the base is 8 × 12 = 96 in2 . Solve 96 × h = 480 to get h = 5 in.

  Example: You have two pitchers that are both in the shape of a cylinder. One of the pitchers has a radius of 4 in. and a height of 8 in. The other pitcher has a radius of 3 in. and a height of 15 in. Which pitcher has the greater capacity?

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  eBook - ePub

  Mathematics for Engineering

  • W Bolton(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  cone is used.

  Figure 2.17 Pyramids

  The term right pyramid is used when the apex is vertically above the centre of the base, otherwise the term oblique pyramid is used. If we think of a right pyramid as built up from a number of layers, then sliding successive layers over another generates an oblique pyramid will the same volume (Figure 2.18 ). A right pyramid and an oblique pyramid will have the same volumes if their base areas are the same and their vertical heights equal.

  Figure 2.18 Right and oblique pyramids

  The volume V of any pyramid is:

  V = × area of base × perrpendicular height

  This can be demonstrated by first considering a triangular-based prism (Figure 2.19 ). Diagonal planes can be used to divide the prism into three equal volume triangular-based pyramids ABCE, DEFC and BCDE. Thus, since the volume of the prism is the base area multiplied by the perpendicular height, the volume of the triangular prism is one third the base area multiplied by the perpendicular height. This argument applies to other prisms.

  Figure 2.19 A prism divided into pyramids

  The total surface area of a pyramid is the sum of the areas of the triangles forming the sides plus the area of the base.

  A cone with a base of radius r and vertical height h (Figure 2.17(c) ) has a base area of πr2 and thus the volume V of a cone is:

  V = πr2 h

  The area of the curved surface of a cone may be obtained by imagining the cone surface to be a sheet of paper which is cut along the line OA and then unrolled onto a flat surface (Figure 2.20 ). The sheet is a segment of a circle with a radius equal to the slant height l of the cone and segment having a circumference equal to that of the cone, i.e. 2πr. The area of a circle of radius l is πl2 and its circumference is 2πl; thus the area of a segment with an arc length 2πr is (2πr/2πl) × πl2 = πrl

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  eBook - ePub

  Science and Mathematics for Engineering

  • 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 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

  14.2.1   Cuboid or rectangular prism

  Science and Mathematics for Engineering. 978-0-367-20475-4, © John Bird. Published by Taylor & Francis. All rights reserved.

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  eBook - ePub

  Mathematics Content for Elementary Teachers

  • 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

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