Mathematics
Surface Area of Sphere
The surface area of a sphere is the total area of the outer surface of the sphere. It is calculated using the formula 4πr^2, where r is the radius of the sphere. This formula represents the sum of all the individual areas of the infinitesimally small pieces that make up the surface of the sphere.
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4 Key excerpts on "Surface Area of Sphere"
eBook - PDF- Charles P. McKeague, Kate Duffy Pawlik(Authors)
- 2014(Publication Date)
- XYZ Textbooks(Publisher)
In this section we will learn how to compute the surface area of any sphere, such as a planet, given its radius. We will also find the surface area of other three-dimensional shapes, such as cubes and cylinders. Surface Area of a Rectangular Solid The figure below shows a closed box with length l, width w, and height h. The sur- faces of the box are labeled as sides, top, bottom, front, and back. A box like this is called a rectangular solid. In general, a rectangular solid is a closed figure in which all sides are rectangular that meet at right angles. To find the surface area of the box, we add the areas of each of the six surfaces that are labeled in the figure. Surface area = side + side + front + back + top + bottom S = l ⋅ h + l ⋅ h + h ⋅ w + h ⋅ w + l ⋅ w + l ⋅ w = 2lh + 2hw + 2lw NASA h l Base Back w Top Front Side Side A box with dimensions l, w, and h S = 2lh + 2hw + 2lw Surface Area 536 Chapter 8 Geometry Find the surface area of the box shown here. Solution To find the surface area we find the area of each surface individually, and then we add them together: Surface area = 2(4 in.)(5 in.) + 2(3 in.)(5 in.) + 2(3 in.)(4 in.) = 40 in 2 + 30 in 2 + 24 in 2 = 94 in 2 The total surface area is 94 square inches. Surface Area of a Cylinder Here are the formulas for the surface area of some right circular cylinders. A right cylinder is a cylinder whose base is a circle and whose sides are perpendicular to the base. The drinking straw shown below has a radius of 0.125 inch and a length of 6 inches. How much material was used to make the straw? Solution Since a straw is a cylinder that is open at both ends, we find the amount of material needed to make the straw by calculating the surface area. S = 2πrh ≈ 2(3.14)(0.125)(6) = 4.71 in 2 It takes about 4.71 square inches of material to make the straw. VIDEO EXAMPLES SECTION 8.3 Example 1 5 in. 4 in. 3 in. Note Saying the sides of a cylinder are perpendicular to the base means that they meet at right angles.
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 - 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)
Also the curved surface area of the circumscribing cylinder is 27rR x 2R which equals 47rR 2 . But the surface area of the sphere is 47rR 2 and so the surface area of the sphere is equal to the curved surface area of the circumscribing cylinder. Ex. A sphere has diameter 4 in. Find the volume correct to the nearest cubic inch and the surface area correct to the nearest square inch. V = ^TTD 3 = i x 3-1416 x 4 3 6 = 33-5104 Volume = 34 in 3 (correct to the nearest cubic inch) A = 7 r D 2 = 3-1416 x 4 2 = 50-2656 Area = 50 in 2 (correct to the nearest square inch) Ex. Find correct to the nearest cubic inch the volume of a sphere of diameter 5 in. Find also the surface area correct to the nearest square inch. AREAS OF SIMILAR FIGURES We must first realise that any flat area can be covered with squares. Only a few squares are shown in Diagram 29. As we increase the number of squares the uncovered area becomes as small as we please. 98 M A T H E M A T I C S F O R E N G I N E E R S DIAGRAM 2 9 Consider now the following diagram which shows two SIMILAR FIGURES. DIAGRAM 3 0 The figure on the right is a magnified form of the figure on the left. The magnification is 3 to 1 which means that the right-hand figure is 3 times as big in all directions as the figure on the left. To be more precise it means that if we draw any straight line in the left-hand figure then the corresponding straight line in the right-hand figure is 3 times as long. These straight lines are called M E N S U R A T I O N 99 CORRESPONDING DIMENSIONS. If we draw the squares on corresponding dimensions then the right-hand square will have 9 times the area of the left-hand square. The ratio of the areas of the squares drawn on corresponding dimensions is 9 to I. Now the left-hand figure can be covered with squares. To each of these squares there corresponds a square in the right-hand figure having 9 times the area and so The ratio of the areas of the two figures is 9 to 1.
eBook - PDFGeometry
A Self-Teaching Guide
- Steve Slavin, Ginny Crisonino(Authors)
- 2004(Publication Date)
- Wiley(Publisher)
Volume of a hemisphere formula V = 1 2 4 3 πr 3 , which is 2 3 πr 3 , so V = 2 3 πr 3 Example 16: Find the volume of a hemisphere with a radius of 10 inches. Solution: V = πr 3 = (0.67)(3.14)(10) 3 = (2.1038)(1,000) = 2,103.8 cubic inches Example 17: Find the volume of a sphere with a diameter of 30 inches. Solution: The formula for the volume of a sphere uses the radius, not the diameter, of the sphere. We’ll find the radius by dividing the diameter by 2. r = = = 15 Volume = πr 3 = (1.33)(3.14)(15) 3 = (4.1762)(3,375) = 14,094.7 cubic inches 4 3 30 2 d 2 2 3 4 3 sphere hemisphere 178 GEOMETRY Example 18: Find the volume of a hemisphere with a radius of 4 feet. Solution: V = πr 3 = (.67)(3.14)(4) 3 = (2.1038)(64) = 134.643 cubic feet We’ve looked at the surface area of other three-dimensional solids, so why not a sphere? Surface area of a sphere formula SA = 4πr 2 Example 19: Find the surface area of a sphere with a radius of 5 yards. Solution: SA = 4πr 2 = 4(3.14)(5) 2 = 4(3.14)(25) = 314 square yards Example 20: Find the surface area of a sphere with a diameter of 12.2 yards. Solution: First we have to find the radius of the sphere. r = 2 d = 12 2 .2 = 6.1 SA = 4πr 2 = 4(3.14)(6.1) 2 = (12.56)(37.21) = 467.358 square yards Can you figure out the formula for the surface area of a hemisphere? (Don’t worry about the flat part; just get the spherical part.) Surface area of a hemisphere formula SA = (4πr 2 ) = 2πr 2 , or SA = 2πr 2 Example 21: Find the surface area of a hemisphere with a radius of 2.5 feet. Solution: SA = 2πr 2 = 2(3.14)(2.5) 2 = (6.28)(6.25) = 39.25 square feet 1 2 2 3 Volume and Surface Area of Three-dimensional Polygons 179 SELF-TEST 3 1. Find the volume for a sphere with a radius of a. 2 feet b. 9 inches c. 3.5 yards 2. Find the volume for a sphere with a diameter of 15 inches. 3. Find the volume of a hemisphere with a radius of 18 feet. 4. Find the surface area of a sphere with a radius of a.
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