Mathematics
Surface of Pyramids
The surface area of a pyramid is the total area of all its faces, including the base and lateral faces. To find the surface area, one can use the formula: surface area = base area + (0.5 × perimeter of base × slant height). This measurement is important in geometry and engineering for determining the amount of material needed to cover the pyramid.
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4 Key excerpts on "Surface of Pyramids"
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
This is the volume, the quantity that the three-dimensional object holds. Suppose we can fit 10 sugar cubes across the length of the bottom of the box, and 4 cubes along the width of the bottom of the box and 6 cubes high. Then we could fit a total of 10 3 4 3 6 5 240 cubes. This sugar cube example helps us to see why volume is measured in cubic units (because we would fill it with cubes). INVESTIGATION 9.3a Building a Box and Filling a Box Explorations Manual 9.14 UNDERSTANDING THE SURFACE AREA OF PRISMS AND PYRAMIDS Very simply, the surface area of a prism or pyramid is equal to the sum of the areas of all of its faces. Thus, to determine the surface area, we need to be able to determine the dimensions of each face. Let us consider a rectangular prism first. Determine the surface area of the rectangular prism in Figure 9.19 and then read on. It may help first to draw a net of that prism, which is shown at the right. The Common Core State Standards recom-mends using nets extensively in sixth grade to help students understand surface area. These visuals help students of all ages to better understand the concept. 12 4 6 Figure 9.19 There are patterns in this prism that make its surface area easier to determine. You can see that the bottom and top bases are congruent; similarly, the front and back faces are congruent, and the right and left faces are congruent. Thus, S ur f a ce ar e a of the pr is m 5 2 H20849 24 in 2 H20850 1 2 H20849 7 2 in 2 H20850 1 2 H20849 4 8 in 2 H20850 5 2 88 in 2 With a concrete example under our belts, let us now investigate the surface area of prisms and pyramids whose bases are regular polygons. What do we know about the different surfaces of prisms and pyramids that would make the task easier? Look at the diagrams in Figure 9.20, showing three prisms, one pyramid, and their nets. What do you see? Think before reading on. . . . Copyright 2016 Cengage Learning.
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)
Suppose that the lateral faces of the pyramid in Exercise 5 have , , , and . If each side of the square base measures 4 in., find the total surface area of the pyramid. 10. Suppose that the base of the hexagonal pyramid in Exercise 6 has an area of 41.6 cm and that each lateral face has an area of 20 cm . Find the total (surface) area of the pyramid. 11. Suppose that the base of the square pyramid in Exercise 5 has an area of 16 cm and that the altitude of the pyramid measures 6 cm. Find the volume of the square pyramid. 12. Suppose that the base of the hexagonal pyramid in Exercise 6 has an area of 41.6 cm and that the altitude of the pyramid measures 3.7 cm. Find the volume of the hexagonal pyramid. 13. Assume that the number of sides in the base of a pyramid is n . Generalize the results found in earlier exercises by answering each of the following questions. a) What is the number of vertices? b) What is the number of lateral edges? c) What is the number of base edges? d) What is the total number of edges? e) What is the number of lateral faces? f) What is the total number of faces? (Note: Lateral faces and base faces.) 14. Refer to the prisms of Exercises 1 and 2. Which of these have symmetry with respect to one (or more) plane(s)? 15. Refer to the pyramids of Exercises 3 and 4. Which of these have symmetry with respect to one (or more) plane(s)? 16. Refer to the prisms of Exercises 1 and 2. Which of these prisms have symmetry with respect to a point? 17. Refer to the pyramids of Exercises 3 and 4. Which of these pyramids have symmetry with respect to a line? 18. Consider any regular pyramid. Indicate which line segment has the greater length: a) Slant height or altitude? b) Lateral edge or radius of the base? 19.
eBook - PDFMathematics
A Practical Odyssey
- David Johnson, , Thomas Mowry, , David Johnson, Thomas Mowry(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 572 CHAPTER 8 Geometry Because of its near perfection, some people believe that the pyramid contains mystic puzzles and the answers to the riddles of the universe. For instance, if you divide the semiperimeter of the base (the distance halfway around, or two times the length of a side) by the height of the pyramid, an interesting result is obtained: semiperimeter height 5 2 s 755.78 d 481.2 5 3.141230258 . . . The comparison of this number to the modern-day approximation of p is remarkable ( p is approximately 3.141592654 . . .). Was this planned, or is it merely coincidental? The debate continues. As surveyors, astronomers, and architects, the early Egyptians were un-doubtedly concerned with the mathematics of a circle. By taking careful mea-surements, they knew that the circumference of a circle was proportional to its diameter—that is, that circumference/diameter 5 constant. Today, we call this constant p . On the basis of a contemporary interpretation of an ancient docu-ment known as the Rhind Papyrus, it appears that the Egyptians would have concluded that p 5 256 81 ( < 3.16). n The Rhind Papyrus Most of our knowledge of early Egyptian mathematics was obtained from two famous papyri: the Moscow (or Golenischev) Papyrus and the Rhind Papyrus. In 1858, the Scottish lawyer and amateur archeologist Henry Rhind was vacationing in Luxor, Egypt. While investigating the buildings and tombs of Thebes, he came across an old rolled-up papyrus. The papyrus was written by the scribe Ahmes around 1650 B . C . and contained eighty-four mathemati-cal problems and their solutions. We know that the papyrus is a copy of an older original, for it begins: “The entrance into the knowledge of all existing things and all obscure secrets.
eBook - PDFGeometry
A Self-Teaching Guide
- Steve Slavin, Ginny Crisonino(Authors)
- 2004(Publication Date)
- Wiley(Publisher)
Once we have all this information, we’ll add these amounts to find the surface area of the entire rectangular solid. a. Surface area of top = lw = (6 inches)(4 inches) = 24 square inches Surface area of front = lh = (6 inches)(2 inches) = 12 square inches Surface area of side = wh = (4 inches)(2 inches) = 8 square inches Surface area = 2(24) + 2(12) + 2(8) = 88 square inches b. Surface area of top = lw = (18 feet)(9 feet) = 162 square feet Surface area of front = lh = (18 feet)(6 feet) = 108 square feet Surface area of side = wh = (9 feet)(6 feet) = 54 square feet Surface area = 2(162 square feet) + 2(108 square feet) + 2(54 square feet) = 324 square feet + 216 square feet + 108 square feet = 648 square feet Pyramids The world’s most famous pyramids are, of course, in Egypt. Even though we may be thousands of miles away, if we know some of their dimensions, we can still cal- culate their impressive volume and surface area. First a general definition: A pyramid is a geometric solid having any polygon as one face, where all the other faces are triangles meeting at a common vertex. The pyramid is named after the polygon forming the face from which the triangles start. To keep things as simple as possible, the face of all the pyramids we’ll consider will be its base. In addition, we’ll work with pyramids having just three bases: the Volume and Surface Area of Three-dimensional Polygons 173 triangle-based pyramid (tetrahedron), the rectangle-based pyramid, and the right square–based pyramid. Volume of a pyramid formula is V = 1 3 Bh where B is the area of the base and h is the height of the pyramid. Find the volume of a pyramid with a base area of 10 and a height of 15. Volume = Bh = (10)(15) = 50 Example 11: Find the volume of the following pyramid. Solution: This pyramid has a base that’s a square. We know this because the length and width are equal. B is the area of the base, which is (7)(7). The height is 6.
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