3-D Shapes Unit 7: Lesson 11 Surface Area and Volume of Spheres Holt Geometry Texas ©2007

Objectives and Student Expectations TEKS: G2B, G3B, G6B, G8D, G11D The student will make conjectures about 3-D figures and determine the validity using a variety of approaches. The student will construct and justify statements about geometric figures and their properties. The student will use nets to represent and construct 3-D figures. The student will find surface area and volume of prisms, cylinders, cones, pyramids, spheres, and composite figures. The student will describe the effect on perimeter, area, and volume when one or more dimensions of a figure are changed.

A sphere is the locus of points in space that are a fixed distance from a given point called the center of a sphere. A radius of a sphere connects the center of the sphere to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres

Example: 1 Find the volume of the sphere. Give your answer in terms of . Volume of a sphere. V = 2304 in3 Simplify.

Example: 2 Find the diameter of a sphere with volume 36,000 cm3. Volume of a sphere. Substitute 36,000 for V. 27,000 = r3 r = 30 Take the cube root of both sides. d = 60 cm d = 2r

Example: 3 Find the volume of the hemisphere. Volume of a hemisphere Substitute 15 for r. V = 2250 m3 Simplify.

Example: 4 Volume of a sphere Substitute for V. r = 12 ft Simplify. Find the radius of a sphere with volume 2304 ft3. Volume of a sphere Substitute for V. r = 12 ft Simplify.

Example: 5 A sporting goods store sells exercise balls in two sizes, standard (22-in. diameter) and jumbo (34-in. diameter). How many times as great is the volume of a jumbo ball as the volume of a standard ball? standard ball: jumbo ball: A jumbo ball is about 3.7 times as great in volume as a standard ball. Picture Source : http://simple.wikipedia.org

Example: 6 A hummingbird eyeball has a diameter of approximately 0.6 cm. How many times as great is the volume of a human eyeball with a diameter of 2.5 cm as the volume of a hummingbird eyeball? hummingbird: human: The human eyeball is about 72.3 times as great in volume as a hummingbird eyeball. Picture Source : allisonians.blogspot.com

A baseball is made of two pieces of leather as shown to the left A baseball is made of two pieces of leather as shown to the left. If you cut up the middle section resembling a rectangle, you can make 4 perfect circles. Therefore the formula for the surface area of a baseball is the area of 4 circles.

Example: 7 Find the surface area of a sphere with diameter 76 cm. Give your answers in terms of . S = 4r2 Surface area of a sphere S = 4(38)2 S = 5776 cm2

Example: 8 Find the volume of a sphere with surface area 324 in2. Give your answers in terms of . S = 4r2 Surface area of a sphere 324 = 4r2 Substitute 324 for S. r = 9 Solve for r. in3 Substitute 9 for r. The volume of the sphere is 972 in3.

Example: 9 Find the surface area of a sphere with a great circle that has an area of 49 mi2. Area of a circle A = r2 Substitute 49 for A. 49 = r2 Solve for r. r = 7 S = 4r2 Substitute 7 for r. = 4(7)2 = 196 mi2

Example: 10 Find the surface area of the sphere. S = 4r2 Surface area of a sphere S = 4(25)2 Substitute 25 for r. S = 2500 cm2

Example: 11 The radius of the sphere is multiplied by . Describe the effect on the volume. radius multiplied by : original dimensions: Notice that . If the radius is multiplied by , the volume is multiplied by , or .

Example: 12 The radius of the sphere is divided by 3. Describe the effect on the surface area. original dimensions: dimensions divided by 3: S = 4r2 S = 4r2 S = 4(3)2 = 36 m3 S = 4(1)2 = 4 m3 The surface area is divided by 9.

Example: 13 Find the surface area and volume of the composite figure. Give your answer in terms of . Step 1 Find the surface area of the composite figure. The surface area of the composite figure is the sum of the curved surface area of the hemisphere, the lateral area of the cylinder, and the base area of the cylinder.

Example: 13 continued Find the surface area and volume of the composite figure. Give your answer in terms of . L(cylinder) = 2rh = 2(6)(9) = 108 in2 B (cylinder) = r2 = (6)2 = 36 in2 The surface area of the composite figure is S = 72 + 108 + 36 = 216 in2.

Example: 13 continued Find the surface area and volume of the composite figure. Give your answer in terms of . Step 2 Find the volume of the composite figure. The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the cylinder. The volume of the composite figure is 144 + 324 = 468 in3.

Example: 14 Find the surface area and volume of the composite figure (an empty hemisphere inscribed in a cylinder). Step 1 Find the surface area of the composite figure. The surface area of the composite figure is the sum of the curved surface area of the hemisphere, the lateral area of the cylinder, and the base area of the cylinder.

Example: 14 continued Find the surface area and volume of the composite figure. L(cylinder) = 2rh = 2(3)(5) = 30 ft2 B(cylinder) = r2 = (3)2 = 9 ft2 The surface area of the composite figure is S = 18 + 30 + 9 = 57 ft2.

Example: 14 continued Find the surface area and volume of the composite figure. Step 2 Find the volume of the composite figure. The volume of the composite figure is the volume of the cylinder minus the volume of the hemisphere. V = 45 – 18 = 27 ft3