11.8 Vocabulary sphere center of a sphere radius of a sphere hemisphere great circle

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

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. The height of the hemisphere is equal to the radius.

V(hemisphere) = V(cylinder) – V(cone) The volume of a sphere with radius r is twice the volume of the hemisphere, or .

Find the volume of the sphere. Give your answer in terms of .

Example 1B: Finding Volumes of Spheres Find the diameter of a sphere with volume 36,000 cm3. Find the volume of the hemisphere.

Example 2: Sports Application A sporting goods store sells exercise balls in two sizes, standard (12-in. diameter) and jumbo (24-in. diameter). How many times as great is the volume of a jumbo ball as the volume of 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.

4r2 ≈ 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 4r2 ≈ S. As the number of pyramids increases, the approximation gets closer to the actual surface area.

Find the surface area of a sphere with diameter 76 cm Find the surface area of a sphere with diameter 76 cm. Give your answers in terms of .

Example 3B: Finding Surface Area of Spheres Find the volume of a sphere with surface area 324 in2. Give your answers in terms of . Find the surface area of a sphere with a great circle that has an area of 49 mi2.

Example 4: Exploring Effects of Changing Dimensions The radius of the sphere is multiplied by . Describe the effect on the volume. Why? The radius of the sphere is divided by 3. Describe the effect on the surface area.

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 .

Check It Out! Example 5 Find the surface area and volume of the composite figure.

Lesson Quiz: Part I Find each measurement. Give your answers in terms of . 1. the volume and surface area of the sphere 2. the volume and surface area of a sphere with great circle area 36 in2 3. the volume and surface area of the hemisphere V = 36 cm3; S = 36 cm2 V = 288 in3; S = 144 in2 V = 23,958 ft3; S = 3267 ft2

Lesson Quiz: Part II 4. A sphere has radius 4. If the radius is multiplied by 5, describe what happens to the surface area. 5. Find the volume and surface area of the composite figure. Give your answer in terms of . The surface area is multiplied by 25. V = 522 ft3; S = 267 ft2