Geometry Spheres CONFIDENTIAL

Warm Up Describe the effect on the volume that results from the given change. The side length of a cube are multiplied by ¾. The height and the base area of a prism are multiplied by 5. 1) the volume is decreased by 27/64 times. 2) the volume is increased by 25 times. CONFIDENTIAL

Spheres 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 to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres. Center Hemisphere Great circle Radius Next Page: CONFIDENTIAL

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. h r h r Next Page: CONFIDENTIAL

h r h r CONFIDENTIAL

Volume of a Sphere r CONFIDENTIAL

Finding Volumes of Spheres Find each measurement. Give your answer in terms of . A) The volume of the sphere 9 cm Next Page: CONFIDENTIAL

B) The diameter of a sphere with volume 972 in 3 9 cm Next Page: CONFIDENTIAL

C) The volume of the hemisphere CONFIDENTIAL

1) Find the radius of a sphere with volume 2304 ft . Now you try! 1) Find the radius of a sphere with volume 2304 ft . 3 1) 11.7 ft CONFIDENTIAL

Biology Application Giant squid need large eyes to see their prey in low light. The eyeball of a giant squid is approximate a sphere with a diameter of 25 cm, which is bigger than a soccer ball. A human eyeball is approximate a sphere with a diameter of 2.5 cm. How many times as great is the volume of a giant squid eyeball as the volume of a human eyeball? A giant squid eyeball is about 1000 times as great in volume as a human eyeball. CONFIDENTIAL

Now you try! 2) A hummingbird eyeball has a diameter of approximately 0.625 cm. How many times as great is the volume of a human eyeball as the volume of a hummingbird eyeball. A human eyeball is approximate a sphere with a diameter of 2.5 cm. ? 2) the volume of human eye ball is 64 times the volume of humming bird. CONFIDENTIAL

In the figure, the vertex of the pyramid is at the center of the sphere. The height of the pyramid is approximate the radius r of the sphere. Suppose the entire sphere is filled with n pyramids that each have base area B and height r. Next Page: CONFIDENTIAL

Next Page: CONFIDENTIAL

If the pyramids fill the sphere, the total area of the bases is approximate equal to the surface area of the sphere S, so 4 r = S. As the number of pyramids increases, the approximation gets closer to the actual surface area. 2 CONFIDENTIAL

Surface Area of a Sphere The surface area of a sphere with radius r is S = 4 r . 2 r CONFIDENTIAL

Finding Surface Area of Spheres Find each measurement. Give your answers in terms of . Next Page: CONFIDENTIAL

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3) Find the surface area of the sphere. Now you try! 3) Find the surface area of the sphere. 50 cm 3) 2500∏ cm2 CONFIDENTIAL

Exploring Effects of Changing Dimensions The radius of the sphere is tripled. Describe the effect on the volume. 3 m CONFIDENTIAL

4) The radius of the sphere above is divided by 3. Now you try! 4) The radius of the sphere above is divided by 3. Describe the effect on the surface area. 4) the volume decrease by 9 times. CONFIDENTIAL

Finding Surface Areas and Volumes of Composite Figures Find the surface area and volume of the composite figure. Give your answers in terms of . 7 cm 25 cm Step 1 Find the surface area of the composite figure. The surface area of the composite figure is the sum of the surface area of the hemisphere and the lateral area of the cone. Next Page: CONFIDENTIAL

Step 2 Find the volume of the composite figure. First find the height of the cone. 7 cm 25 cm The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the cone. CONFIDENTIAL

5) Find the surface area and volume of the composite figure. Now you try! 5) Find the surface area and volume of the composite figure. 3 ft 5 ft 5) 57∏ ft2 CONFIDENTIAL

Now some problems for you to practice ! CONFIDENTIAL

Assessment 1)Find each measurement. Give your answers in terms of . The volume of the hemisphere B) The volume of the sphere 11 in. 1 m 1a) 887.33 ∏ in3 1b) 1.33 ∏ m3. CONFIDENTIAL

2)Approximately how many times as great is the volume of the grapefruit as the volume of the lime? 10 cm 5 cm 2) 8 times CONFIDENTIAL

3)Find each measurement. Give your answers in terms of . The surface area of the sphere B) The surface area of the sphere 16 yd 3a) 256∏ yd2 3b) 196∏ cm2. CONFIDENTIAL

4) Describe the effect of each change on the given measurement of the figure. Surface area The dimensions are doubled. B) Volume The dimensions are multiplied by ¼. 15 in. 16 cm 4a) Increases by 4 times 4b) Decreases by 64 times. CONFIDENTIAL

5) Find the surface area and volume of the composite figure. 5a) SA = 36∏ ft3 V = 30.67∏ ft3 CONFIDENTIAL

Let’s review CONFIDENTIAL

Spheres 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 to any point on the sphere. A hemisphere is half of a sphere. A great circle divides a sphere into two hemispheres. Center Hemisphere Great circle Next Page: CONFIDENTIAL

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. h r h r Next Page: CONFIDENTIAL

h r h r CONFIDENTIAL

Volume of a Sphere r CONFIDENTIAL

Finding Volumes of Spheres Find each measurement. Give your answer in terms of . A) The volume of the sphere 9 cm Next Page: CONFIDENTIAL

B) The diameter of a sphere with volume 972 in 3 9 cm Next Page: CONFIDENTIAL

C) The volume of the hemisphere CONFIDENTIAL

Biology Application Giant squid need large eyes to see their prey in low light. The eyeball of a giant squid is approximate a sphere with a diameter of 25 cm, which is bigger than a soccer ball. A human eyeball is approximate a sphere with a diameter of 2.5 cm. How many times as great is the volume of a giant squid eyeball as the volume of a human eyeball? A giant squid eyeball is about 1000 times as great in volume as a human eyeball. CONFIDENTIAL

In the figure, the vertex of the pyramid is at the center of the sphere. The height of the pyramid is approximate the radius r of the sphere. Suppose the entire sphere is filled with n pyramids that each have base area B and height r. Next Page: CONFIDENTIAL

Next Page: CONFIDENTIAL

If the pyramids fill the sphere, the total area of the bases is approximate equal to the surface area of the sphere S, so 4 r = S. As the number of pyramids increases, the approximation gets closer to the actual surface area. 2 CONFIDENTIAL

Surface Area of a Sphere The surface area of a sphere with radius r is S = 4 r . 2 r CONFIDENTIAL

Finding Surface Area of Spheres Find each measurement. Give your answers in terms of . Next Page: CONFIDENTIAL

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Exploring Effects of Changing Dimensions The radius of the sphere is tripled. Describe the effect on the volume. 3 m CONFIDENTIAL

Finding Surface Areas and Volumes of Composite Figures Find the surface area and volume of the composite figure. Give your answers in terms of . 7 cm 25 cm Step 1 Find the surface area of the composite figure. The surface area of the composite figure is the sum of the surface area of the hemisphere and the lateral area of the cone. Next Page: CONFIDENTIAL

Step 2 Find the volume of the composite figure. First find the height of the cone. 7 cm 25 cm The volume of the composite figure is the sum of the volume of the hemisphere and the volume of the cone. CONFIDENTIAL

You did a great job today! CONFIDENTIAL