Mathematics
Surface Area of Cylinder
The surface area of a cylinder is the total area of all its surfaces, including the two circular bases and the curved surface. To calculate the surface area, one can use the formula 2πr² + 2πrh, where r is the radius of the base and h is the height of the cylinder. This measurement is important in various real-world applications and mathematical problems.
Written by Perlego with AI-assistance
Related key terms
1 of 5
10 Key excerpts on "Surface Area of Cylinder"
No longer available |Learn more- Tom Bassarear, Meg Moss(Authors)
- 2015(Publication Date)
- Cengage Learning EMEA(Publisher)
. . . 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. (Use an empty toilet paper tube or paper towel tube and cut it open, or wrap a piece of paper around a soup can or soda can.) What is the shape of the lateral face? Do this yourself before reading on. . . . h r Figure 9.21 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. 542 CHAPTER 9 Geometry as Measurement Unless otherwise noted, all content on this page is © Cengage Learning Yes, it is a rectangle! The height of the rectangle is equal to the height of the cylinder. What about the length of the base of the rectangle? How is it connected to the cylinder’s base? Think and then read on. . . . The length of the base of the rectangle is equal to the circumference of the circle (Figure 9.22)! With this knowledge, see if you can now discover the formula for the surface area of any cylinder in terms of the radius of the cylinder and the height of the cylinder. Then read on. . . . h h r Circumference of circle = 2 πr lateral face Since the Common Core State Standards stress thinking about nets, what would the net of a cylinder look like? Try sketching it and then read on. Base Base Curved Surface • The area of the base is p r 2 , and because we have two bases, the area of the two bases is 2 p r 2 .
eBook - PDFMathematics for the General Course in Engineering
The Commonwealth and International Library: Mechanical Engineering Division, Volume 2
- John C Moore, N. Hiller, G. E. Walker(Authors)
- 2013(Publication Date)
- Pergamon(Publisher)
To find the area of a rectangle we multiply the length by the breadth and so AREA OF CURVED SURFACE OF A CYLINDER = CIRCUMFERENCE x HEIGHT If the diameter of the base circle is D then the base circumference is 7rD. We can write shortly A -TTDH Note the following alternative. If the radius of the base circle is R then the base circumference is InR. We can write shortly A -2TTRH Ex. Find correct to the nearest cubic inch the volume of a right circular cylinder of base diameter 9 in and height 6 in. Find also the curved surface area correct to the nearest square inch. D = 9 and H = 6 V = | 7 r D 2 H = - x 3-1416 x 9 2 x 6 4 - 381-7044 Volume = 382 in 3 (correct to the nearest cubic inch) A = T T DH - 3-1416 x 9 x 6 = 169-6464 Curved surface area = 170 in 2 (correct to the nearest square inch) M E N S U R A T I O N 79 E x . Find correct to the nearest cubic inch the volume of a right circular cylinder of base radius 3-6 in and height 5«2 in. Find also the total surface area correct to the nearest square inch. T H E R I G H T C I R C U L A R C O N E T h e following diagram shows a solid k n o w n as a R I G H T C I R C U L A R C O N E . D DIAGRAM 15 T h e cone is called C I R C U L A R because the base is a C I R C L E and it is called R I G H T because the line in the diagram joining the t o p of the cone to the centre of the base circle is at R I G H T angles to the base. Volume of a Cone Obtaining the formula for the volume of a right circular cone presents quite a problem because the only quick a p p r o a c h involves advanced mathematics. If we restrict ourselves to easy mathematics then it is still possible to produce the formula if we are willing to use the m e t h o d of successive approximations. This extremely i m p o r t a n t method is used in m a n y branches of m a t h e -matics. It is easy to apply but requires patience.
eBook - PDFMathematical Practices, Mathematics for Teachers
Activities, Models, and Real-Life Examples
- Ron Larson, Robyn Silbey(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Find the circumference of a circle. Find the area of a circle. 12.2 Surface Areas of Circular Solids (page 469) 27–44 Find the surface area of a right circular cylinder. Find the surface area of a right circular cone. Find the surface area of a sphere. 12.3 Volumes of Circular Solids (page 479) 45–58 Find the volume of a cylinder. Find the volume of a cone. Find the volume of a sphere. Important Concepts and Formulas circle (p. 459) center of a circle (p. 459) radius (p. 459) diameter (p. 459) π (pi) (p. 460) circumference (p. 460) circular arc (p. 461) area of a circle (p. 462) sector of a circle (p. 463) right circular cylinder (p. 469) lateral surface of a right circular cylinder (p. 469) surface area of a right circular cylinder (p. 469) surface area of a right circular cone (p. 471) slant height of a right circular cone (p. 471) sphere (p. 473) center of a sphere (p. 473) surface area of a sphere (p. 473) cross section (p. 473) volume of a cylinder (p. 479) volume of a cone (p. 481) volume of a sphere (p. 483) Circles The following relationships are true for a circle with radius r, diameter d, circumference C, and area A. • d = 2r or r = d — 2 C d r • π = C — d • C = πd or C = 2πr • A = πr 2 Cylinders The surface area S and volume V of a right circular cylinder with area of base B, radius of base r, and height h are shown. h r • S = 2πr 2 + 2πrh • V = Bh or V = πr 2 h Cones The surface area S and volume V of a right circular cone with area of base B, radius of base r, height h, and slant height ℓ are shown. • S = πr 2 + πr ℓ h r • V = 1 — 3 Bh or V = 1 — 3 πr 2 h Spheres The surface area S and volume V of a sphere with radius r are shown. • S = 4πr 2 r • V = 4 — 3 πr 3 Monkey Business Images/Shutterstock.com Copyright 2014 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).
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
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. SA 5 LA 1 2 A B where SA 5 surface area LA 5 lateral area A B 5 area of base These examples illustrate the method of computing lateral areas and surface areas of right prisms and right circular cylinders. EXAMPLES 1. A shipping crate is shown in Figure 28–27. a. Compute the lateral area. b. Compute the total surface area. Find the perimeter of the base. P B 5 2 s 6.0 ft d 1 2 s 4.5 ft d 5 21 ft Find the lateral area. a. LA 5 21 ft 3 4.0 ft 5 84 sq ft Ans Find the area of the base. A B 5 6.0 ft 3 4.5 ft 5 27 sq ft Find the surface area. b. SA 5 84 sq ft 1 2 s 27 sq ft d 5 138 sq ft Ans 2. A cylinder hotwater tank has a diameter of 0.625 meter and a height of 1.85 meters. a. Find the lateral area. b. Find the surface area. Find the circumference of the circular base. C B < 3.1416 s 0.625 m d < 1.9635 m Find the lateral area. a. LA < 1.9635 m 3 1.85 m < 3.63 m 2 (rounded) Ans Find the area of the circular base. A B < 3.1416 s 0.3125 m d 2 < 0.3068 m 2 Find the surface area. b. SA < 3.63 m 2 1 2 s 0.3068 m 2 d < 4.24 m 2 Ans 4 9 -0 0 6 9 -0 0 4 9 -6 0 Figure 28–27 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. Unit 28 PRISMS AND CYLINDERS: VOLUMES, SURFACE AREAS, AND WEIGHTS 713 Calculator Method a. LA 5 p 3 .625 3 1.85 5 3.632466506, 3.63 m 2 (rounded) Ans b.
- Frank R. Spellman, Nancy E. Whiting(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
The area of a rectangle is found by multiplying the length ( L ) times width ( W ) (see Figure 2.4). Area = L × W (2.6) ■ EXAMPLE 2.46 Problem: Find the area of the rectangle shown in Figure 2.5. Solution: Area = L × W = 14 ft × 6 ft = 84 ft 2 To find the area of a circle, we need to introduce a new term, the radius , which is represented by r . The circle shown in Figure 2.6 has a radius of 6 ft. The radius is any straight line that radiates from the center of the circle to some point on the circumference. By definition, all radii (plural of radius) of the same circle are equal. The surface area of a circle is determined by multiplying π times the radius squared: A = π × r 2 (2.7) L W FIGURE 2.4 Rectangle. 14 ft 14 ft 6 ft 6 ft FIGURE 2.5 Area of a rectangle for Example 2.46. 65 Basic Math Operations where A = Area. π = pi = 3.14. r = Radius of circle = one half of the diameter. ■ EXAMPLE 2.47 Problem: What is the area of the circle shown in Figure 2.6? Solution: Area of circle = π × r 2 = π × 6 2 = 3.14 × 36 = 113 ft 2 If we are assigned to paint a water storage tank, we must know the surface area of the walls of the tank to determine how much paint is required. In this case, we need to know the area of a circular or cylindrical tank. To determine the surface area of the tank, we need to visualize the cylindrical walls as a rectangle wrapped around a circular base. The area of a rectangle is found by multiplying the length by the width; in the case of a cylinder, the width of the rectangle is the height of the wall, and the length of the rectangle is the distance around the circle (circumference). Thus, the area ( A ) of the side wall of a circular tank is found by multiplying the circumference of the base ( C = π × D ) times the height of the wall ( H ): A = π × D × H (2.8) A = π × 20 ft × 25 ft = 3.14 × 20 ft × 25 ft = 1570 ft 2 To determine the amount of paint needed, remember to add the surface area of the top of the tank, which is 314 ft 2 .
eBook - PDF- Daniel C. Alexander, Geralyn M. Koeberlein(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
The congruent circles are known as the bases of each cylinder. h Figure 9.27 A right circular cylinder is shown in Figure 9.27; however, the parallel planes (such as P and P9 in Figure 9.26) are not pictured. The line segment joining the centers of the two circular bases is known as the axis of the cylinder. For a right circular cylinder, it is necessary that the axis be perpendicular to the planes of the circular bases; in such a case, the length of the altitude h is the length of the axis. Discover Think of a can of green beans as a right circular cylinder. For the cylinder, the lateral surface is the “label” of the can. If the label were sliced downward by a perpendicular line between the planes, removed, and rolled out flat, it would be rectangular in shape. As shown below, that rectangle would have a length equal to the circumference of the circular base and a width equal to the height of the cylinder. Thus, the lateral area is given by A 5 bh, which becomes L 5 Ch, or L 5 2prh. SSG EXS. 1, 2 h Green Beans r h C H11005 2 H9266r Green Beans SURFACE AREA OF A CYLINDER 9.3 ■ Cylinders and Cones 421 The formula for the lateral area of a right circular cylinder (found in the following theorem) should be compared to the formula L 5 hP, the lateral area of a right prism whose base has perimeter P. Theorem 9.3.1 The lateral area L of a right circular cylinder with altitude of length h and circumference C of the base is given by L 5 hC. Alternative Form: The lateral area of the right circular cylinder can be expressed in the form L 5 2prh, where r is the length of the radius of the circular base. Rather than constructing a formal proof of Theorem 9.3.1, consider the Discover activity shown on page 420. Theorem 9.3.2 The total area T of a right circular cylinder with base area B and lateral area L is given by T 5 L 1 2B.
eBook - PDFGeometry
A Self-Teaching Guide
- Steve Slavin, Ginny Crisonino(Authors)
- 2004(Publication Date)
- Wiley(Publisher)
A cylinder is a uniform solid whose base is a circle. The cylinder shown above has a radius of 2 and a height of 5. Volume of a cylinder formula V = πr 2 h where r is the radius and h is the height of the cylinder See if you can find the volume of this cylinder. V = πr 2 h = (3.14)(2) 2 (5) = 62.8 Example 22: Find the volume of a cylinder with a radius of 3.8 feet and a height of 4.6 feet. Solution: Volume = πr 2 h = (3.14)(3.8) 2 (4.6) = (3.14)(14.44)(4.6) = 208.571 cubic feet Example 23: Find the volume of a cylinder with a diameter of 10 inches and a height of 29 inches. Solution: First we’ll begin by finding the radius. r = = = 5 Volume = πr 2 h = (3.14)(5) 2 (29) = (3.14)(25)(29) = 2,276.5 cubic inches 10 2 d 2 5 2 Volume and Surface Area of Three-dimensional Polygons 181 Example 24: Find the surface area of a cylinder with a radius of 12 inches and a height of 5 inches. Solution: SA = 2πrh = 2(3.14)(12)(5) = 6.28(60) = 376.8 square inches Example 25: Find the surface area of a cylinder with a diameter of 9 feet and a height of 13.4 feet. Solution: Start by finding the radius. r = = = 4.5 SA = 2πrh = 2(3.14)(4.5)(13.4) = 6.28(60.3) = 378.684 square feet SELF-TEST 4 1. Find the volume of a cylinder with a a. radius of 12 feet and a height of 9 feet. b. radius of 4.5 yards and a height of 12 yards. c. diameter of 16 inches and a height of 23 inches. 2. Find the surface area of a cylinder with the same dimensions as those in problem 1. ANSWERS 1. a. Volume = πr 2 h = (3.14)(12) 2 (9) = (3.14)(144)(9) = 4,069.44 cubic feet b. Volume = πr 2 h = (3.14)(4.5) 2 (12) = (3.14)(20.25)(12) = 763.02 cubic yards c. Volume = πr 2 h = (3.14)(8) 2 (23) = (3.14)(64)(23) = 4,622.08 cubic inches 2. a. SA = 2πrh = 2(3.14)(12)(9) = (6.28)(108) = 678.24 square feet b. SA = 2πrh = 2(3.14)(4.5)(12) = (6.28)(54) = 339.12 square yards c. SA = 2πrh = 2(3.14)(8)(23) = (6.28)(184) = 1,155.52 cubic inches 9 2 d 2 182 GEOMETRY
eBook - PDF- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
The volume of a cylinder is 400 cm 3 . If its radius is 5.20 cm, find its height. Determine also its curved surface area 14. A cylinder is cast from a rectangular piece of alloy 5 cm by 7 cm by 12 cm. If the length of the cylinder is to be 60 cm, find its diameter 15. Find the volume and the total surface area of a regular hexagonal bar of metal of length 3 m if each side of the hexagon is 6 cm 16. A block of lead 1.5 m by 90 cm by 750 mm is hammered out to make a square sheet 15 mm thick. Determine the dimensions of the square sheet, correct to the nearest centimetre 17. A cylinder is cast from a rectangular piece of alloy 5.20 cm by 6.50 cm by 19.33 cm. If the height of the cylinder is to be 52.0 cm, determine its diameter, correct to the nearest centimetre 18. How much concrete is required for the construction of the path shown in Figure 14.8, if the path is 12 cm thick? 2m 1.2 m 8.5 m 2.4 m 3.1 m Figure 14.8 148 Section I 14.2.4 Pyramids Volume of any pyramid = 1 3 × area of base × perpendicular height A square-based pyramid is shown in Figure 14.9 with base dimension x by x and perpendicular height h. For the square-base pyramid shown, volume = 1 3 x 2 h h x x Figure 14.9 Problem 9. A square pyramid has a perpendicular height of 16 cm. If a side of the base is 6 cm, determine the volume of a pyramid Volume of pyramid = 1 3 × area of base × perpendicular height = 1 3 × (6 × 6) × 16 = 192 cm 3 Problem 10. Determine the volume and the total surface area of the square pyramid shown in Figure 14.10 if its perpendicular height is 12 cm Volume of pyramid = 1 3 × area of base × perpendicular height = 1 3 (5 × 5) × 12 = 100 cm 3 5 cm 5 cm D C E A B Figure 14.10 The total surface area consists of a square base and 4 equal triangles.
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)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. SURFACE AREA OF A SPHERE Characterization 2 The following theorem claims that the surface area of a sphere equals four times the area of a great circle of that sphere. This theorem, which is proved in many calculus textbooks, treats the sphere as a surface of revolution. 422 CHAPTER 9 ■ SURFACES AND SOLIDS Unless otherwise noted, all content on this page is © Cengage Learning. D iscover Suppose that you use scissors to cut out each pattern. (You may want to copy and enlarge this page.) Then glue or tape the indicated tabs (shaded) to form regular polyhedra. Which regular polyhedron is formed in each pattern? ANSWERS (a) (b) (c) (d) (a) Tetrahedron (b) Octahedron (c) Hexahedron (cube) (d) Dodecahedron For the earth, the equator is a great circle that separates the earth into two hemispheres . The sphere has line symmetry about any line containing the center of the sphere. Similarly, the sphere has symmetry about any plane that contains the center of the sphere. THEOREM 9.4.2 The surface area S of a sphere whose radius has length r is given by . S 4 p r 2 EXAMPLE 3 Find the surface area of a sphere whose radius length is Use your calculator to approximate the result. r 7 in. Fruits such as oranges have the shape of a sphere. Geometry in the Real World 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. SOLUTION Then . Although half of a circle is called a semicircle , remember that half of a sphere is generally called a hemisphere .
eBook - PDF- John Peterson, Robert Smith(Authors)
- 2019(Publication Date)
- Cengage Learning EMEA(Publisher)
The formula for computing volumes of right circular cylinders is the same as that of prisms. The volume of a right circular cylinder 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, A B 5 p r 2 , or 0.7854 d 2 (rounded) h 5 height Example 1 Find the volume of a cylinder with a base area of 30.0 square centimeters and a height of 6.0 centimeters. V 5 30.0 cm 2 3 6.0 cm 5 180 cm 3 Ans UNIT 62 VOLUMES OF PRISMS AND CYLINDERS 467 Example 2 A length of pipe is shown in Figure 62-5. Find the volume of metal in the pipe. Find the area of the outside circle. A B < 0.7854 d 2 A B < (0.7854)(4.00 in.) 2 < 12.566371 sq in. Find the area of the hole. A B < (0.7854)(3.40 in.) 2 < 9.079224 sq in. Find the cross-sectional area. 12.566371 sq in. 2 9.079224 sq in. < 3.487147 sq in. Find the volume. V < 3.487147 sq in. 3 50.0 in. < 174 cu in. Ans COMPUTING HEIGHTS AND BASES OF PRISMS AND CYLINDERS The height of a prism or a right circular cylinder can be determined if the base area and volume are known. Also, the base area can be found if the height and volume are known. Substitute the known values in the volume formula and solve for the unknown value. Example 1 Find the height of a spacer block in the shape of a right rectangular prism that has a base area of 68.40 square inches and a volume of 366.60 cubic inches. Substitute values in the formula and solve for h . 366.6 cu in. 5 68.40 sq in. 3 h h < 5.36 in. Ans Example 2 Calculate the diameter of a piston designed to have a volume of 450.0 cubic centimeters and a length (height) of 10.800 centimeters. Find the base area. 450.0 cm 3 5 A B (10.800 cm) A B < 41.6667 cm 2 Find the piston diameter. 41.6667 cm 2 < 0.7854 d 2 d 2 < 53.051566 cm 2 d < 7.284 cm Ans (rounded) 50.0 in. 3.40 in. Inside Diameter 4.00 in. Outside Diameter FIGURE 62-5 APPLICATION Tooling Up 1. Find the volume of a steel shaft that is 18.64 cm long and has a radius of 1.75 cm.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
