To find the surface area and volume of a sphere

To find the surface area and volume of a sphere

Sphere set of all points in space equidistant from a given point called the center. Surface Area Formula: S.A. = 4πr 2 r Volume Formula: V = 4 πr 3 3

Great circle the intersection of a sphere and a plane containing the center of the sphere r The great circle cuts the sphere into two hemispheres. The circumference of a great circle is the circumference of the sphere.

Example 1A: Finding Volumes of Spheres Find the volume of the sphere. Give your answer in terms of. Volume of a sphere. Substitute 24 2 = 2304 in 3 Simplify. = 12 for r.

Example 1B: Finding Volumes of Spheres Find the diameter of a sphere with volume 36,000 cm 3. Volume of a sphere. Substitute 36,000 for V. 27,000 = r 3 r = 30 d = 60 cm Take the cube root of both sides. d = 2r

Example 1C: Finding Volumes of Spheres Find the volume of the hemisphere. Volume of a hemisphere Substitute 15 for r. = 2250 m 3 Simplify.

Example 2: Sports Application 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.

Example 3A: Finding Surface Area of Spheres Find the surface area of a sphere with diameter 76 cm. Give your answers in terms of. S = 4 r 2 Surface area of a sphere S = 4 (38) 2 = 5776 cm 2

Example 3B: Finding Surface Area of Spheres Find the volume of a sphere with surface area 324 in 2. Give your answers in terms of. S = 4 r 2 324 = 4 r 2 Substitute 324 for S. r = 9 Solve for r. Surface area of a sphere 3 Substitute 9 for r. The volume of the sphere is 972 in 2. 3

Example 3C: Finding Surface Area of Spheres Find the surface area of a sphere with a great circle that has an area of 49 mi 2. A = r 2 49 = r 2 Area of a circle Substitute 49 for A. r = 7 Solve for r. S = 4 r 2 = 4 (7) 2 = 196 mi 2 Substitute 7 for r.

Example 4: Exploring Effects of Changing Dimensions The radius of the sphere is multiplied by. Describe the effect on the volume. original dimensions: radius multiplied by : Notice that. If the radius is multiplied by, the volume is multiplied by, or.

Example 5: Finding Surface Areas and Volumes of Composite Figures 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 5 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 in 2 B(cylinder) = r 2 = (6) 2 = 36 in 2 The surface area of the composite figure is 72 + 108 + 36 = 216 in 2.

Example 5 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 in 3.

Lesson Quiz: Part I Find each measurement. Give your answers in terms of. 1. the volume and surface area of the sphere V = 36 cm 3 ; S = 36 cm 2 2. the volume and surface area of a sphere with great circle area 36 in 2 V = 288 in 3 ; S = 144 in 2 3. the volume and surface area of the hemisphere V = 23,958 ft 3 ; S = 3267 ft 2

Lesson Quiz: Part II 4. A sphere has radius 4. If the radius is multiplied by 5, describe what happens to the surface area. The surface area is multiplied by 25. 5. Find the volume and surface area of the composite figure. Give your answer in terms of. V = 522 ft 3 ; S = 267 ft 2

Formulas for spheres: S.A. = 4πr 2 4 V = πr 3 3