Mathematics
Surface Area of Prism
The surface area of a prism is the sum of the areas of all its faces. To calculate the surface area of a prism, you need to find the area of the base and multiply it by the height of the prism. Then, add the areas of the two bases and the areas of the rectangular sides.
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5 Key excerpts on "Surface Area of Prism"
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.
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 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 541 s s s s s s s s s s s s h h h l l s s s s s Figure 9.20 Unfolding these three-dimensional shapes to create the two-dimensional nets takes practice. With many sixth graders, as well as adults, physically unfolding paper models helps develop understanding. Another option is to draw out each face separately, find the areas of each, and add them up. Either way, to find the surface area of a prism or pyramid, we find the area of each face and add them to find the complete surface area. Because we are still talking about area, we are still measuring in square units—in other words we are covering the surface with squares. UNDERSTANDING THE SURFACE AREA OF CYLINDERS How do you think we might determine the surface area of a cylinder with a circular base (Figure 9.21)? Then read on. . . . To find the surface area of a cylinder, we can begin by finding the area of one base and multiplying that by 2. What about the surface area of the rounded part (the part you would hold if this was a soda can)? How do you think we might find that? If you are not sure, find a cylinder.
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.
eBook - PDFMathematics for Elementary Teachers
A Contemporary Approach
- Gary L. Musser, Blake E. Peterson, William F. Burger(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
The cylinder is 1 meter high. If each stripe makes a constant angle of 60° with the vertical axis of the cylinder, how much surface area is covered by the red stripe? Analyzing Student Thinking 21. Carter asserts that if the bases of a prism are hexagons, then the lateral surface area of the prism is six times the area of any face. Is he correct? Explain. 22. When using the formula S r r h = + 2p ( ) , Amberly claims that this is the formula for the volume since the r r , , and h suggest three dimensions. How should you respond? 23. A problem shows a right regular pyramid with the perim- eter of the base and the height from the apex of the pyra- mid to the center of the base given. Tara, who is asked to find the surface area of the pyramid, says that she can’t because she doesn’t know the slant height. How should you respond? 24. Kennedy says that it is impossible for the lateral sur- face area of a right circular cylinder to be equal to the sum of the areas of the bases since the areas of the bases involve p . Is she correct? Explain. 25. Mason says that the area of a sphere doubles if you dou- ble the radius. How should you respond? 26. In making a (two-dimensional) net of a cylinder, Jalen was confused. He thought that since the bases were circles that two of the edges of the lateral surface would be round. He wondered how it could be a rectangle. How would you help Jalen visualize what the net should be? 27. Mark wants to make a tall skinny cone and a short fat cone (without their bases). 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.
eBook - PDF- J Daniels, M Kropman, J Daniels, M Kropman(Authors)
- 2014(Publication Date)
- Future Managers(Publisher)
276 Module 8 • Mensuration and percentages Three-dimensional (3D) Shape Surface area Base area and volume Right prism (trapezium as base) ⊥ h c d a b H 2 trapeziums and 4 rectangles 2 1 1 2 ( a + b ) × ⊥ h 2 + a H + b H + c H + d H ⊥ h c d a b Area = 1 2 sum of //sides × distance between = 1 2 ( a + b ) h V = area of base × height = 1 2 ( a + b ) ⊥ h × H Cylinder h r 2 circles and a rectangle h h 2π r circumference r ✃ Surface area = 2(π r 2 ) + 2π rh = 2π r 2 + 2π rh = 2π r ( r + h ) • common factor ( NB: Depends if cylinder is open or closed) V = area of base × height = π r 2 h When calculating surface area, read carefully to see if any sides are open or need to be left out. Sphere Surface area = 4π r 2 V = 4 3 π r 3 Cone h r r Circle: π r 2 V = 1 3 area of base × ⊥ h = 1 3 π r 2 h Right pyramid h l b Could be a rectangle: = l × b or a triangle or a parallelogram V = 1 3 area of base × ⊥ h = 1 3 lb ⊥ h Depending the shape of the base V pyramid = 1 3 area of base × ⊥ h 277 N1 Mathematics| Hands-On! Examples 1. A rectangular prism measures 26 cm × 12 cm × 8 cm. Calculate: a) its volume b) its total surface area. 2. A sphere has a diameter of 230 mm. Calculate the following. a) the volume b) the surface area c) the cost of filling the sphere with helium at R31,50/ℓ of helium 3. A cylinder open at one end is used as a water tank. The diameter of the cylinder is 14 cm and its height is 22 cm. Calculate the following. a) the volume b) the total surface area 4. Calculate the volume of the right-angled rectangular pyramid below. 0,95 m 1,4 m 2,4 m 5. a) The hemisphere has a diameter of 30 cm. Calculate the total surface area of the closed hemisphere. b) If the diameter of the hemisphere increases by 2 cm, what percentage would the volume increase be? 6. The volume of a right pyramid with a square base is 24 500 cm 3 . Calculate the dimensions of the base if the perpendicular height is 350 cm.
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