CN#7 Objectives. Vocabulary 5/23/ Spheres. I will learn and apply the formulas for the surface area and volume of a sphere.

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Marion Lloyd
6 years ago
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Warm Up #9: The figure shows a hemisphere and a cylinder with a cone removed from its interior. The cross section of the hemisphere is a circle! with radius: r 2 x 2! so its area is:!

and an inner radius of: x! so its area is:!

1 Warm Up #9: The figure shows a hemisphere and a cylinder with a cone removed from its interior. The cross section of the hemisphere is a circle! with radius: r 2 x 2! so its area is:! A = π(r 2 x 2 ) CN#7 Objectives I will learn and apply the formulas for the surface area and volume of a sphere. The cross section of the cylinder with the cone! removed has an outer radius of: r! and an inner radius of: x! so its area is:!a = πr 2 πx 2 = π(r 2 x 2 ) sphere center of a sphere radius of a sphere hemisphere great circle Vocabulary 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 1

V(hemisphere) = V(cylinder) V(cone) The height of the hemisphere is equal to the radius.

2 The figure shows a hemisphere and a cylinder with a cone removed from its interior. The cross sections have the same area at every level, so the volumes are equal by Cavalieri s Principle. V(hemisphere) = V(cylinder) V(cone) The height of the hemisphere is equal to the radius. The volume of a sphere with radius r is twice the volume of the hemisphere, or. Example 1A: Finding Volumes of Spheres Find the volume of the sphere. Volume of a sphere. = 2304π in 3 Simplify. 2

3 Example 1B: Finding Volumes of Spheres Find the diameter of a sphere with volume 36,000π cm 3. Volume of a sphere. Substitute 36,000π for V. 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: 27,000 = r 3 r = 30 d = 60 cm Take the cube root of both sides. d = 2r A jumbo ball is about 3.7 times as great in volume as a standard ball. In the figure, the vertex of the pyramid is at the center of the sphere. The height of the pyramid is approximately the radius r of the sphere. Suppose the entire sphere is filled with n pyramids that each have base area B and height r. 4πr 2 nb If the pyramids fill the sphere, the total area of the bases is approximately equal to the surface area of the sphere S, so 4πr 2 S. As the number of pyramids increases, the approximation gets closer to the actual surface area. 3

4 Example 3A: Finding Surface Area of Spheres Find the surface area of a sphere with diameter 76 cm. Give your answers in terms of π. S = 4πr 2 Surface area of a sphere S = 4π(38) 2 = 5776π cm 2 Example 3B: Finding Surface Area of Spheres Find the volume of a sphere with surface area 324π in 2. Give your answers in terms of π. S = 4πr 2 324π = 4πr 2 Substitute 324π for S. r = 9 Solve for r. Surface area of a sphere 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 : = 4 Substitute 9 for r. 3 π(9)3 = 972π.in 3 The volume of the sphere is 972π in 3. Notice that. If the radius is multiplied by, the volume is multiplied by, or. 4

5 Example 5: Finding Surface Areas and Volumes of Composite Figures Find the surface area and volume of the composite figure. 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. L(cylinder) = 2πrh = 2π(6)(9) = 108π in 2 B(cylinder) = πr 2 = π(6) 2 = 36π in 2 The surface area of the composite figure is 72π + 108π + 36π = 216π in 2. Example 5 Continued Find the surface area and volume of the composite figure. Give your answer in terms of π. 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 144π + 324π = 468π in 3. 5

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