1. Geometry Spheres
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2. 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.
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3. 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
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4. 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
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5. h r h r
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6. Volume of a Sphere r
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7. Finding Volumes of Spheres
Find each measurement. Give your answer in terms of .
A) The volume of the sphere
9 cm
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8. B) The diameter of a sphere with volume 972 in3
9 cm
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9. C) The volume of the hemisphere
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10. 1) Find the radius of a sphere with volume 2304 ft .
Now you try!
1) Find the radius of a sphere with volume ft . 3
1) 11.7 ft
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11. 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.
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12. Now you try!
2) A hummingbird eyeball has a diameter of approximately 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.
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13. 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.
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15. 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
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16. Surface Area of a Sphere
The surface area of a sphere with radius r is S = 4 r . 2 r
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17. Finding Surface Area of Spheres
Find each measurement. Give your answers in terms of .
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19. 3) Find the surface area of the sphere.
Now you try!
3) Find the surface area of the sphere.
50 cm
3) 2500∏ cm2
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20. Exploring Effects of Changing Dimensions
The radius of the sphere is tripled. Describe the effect on the volume.
3 m
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21. 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.
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22. 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.
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23. 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.
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24. 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
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25. Now some problems for you to practice !
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26. 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) ∏ in3
1b) 1.33 ∏ m3.
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27. 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
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28. 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.
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29. 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.
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30. 5) Find the surface area and volume of the composite figure.
5a) SA = 36∏ ft3 V = 30.67∏ ft3
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31. Let’s review
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32. 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
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33. 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:
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34. h r h r
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35. Volume of a Sphere r
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36. Finding Volumes of Spheres
Find each measurement. Give your answer in terms of .
A) The volume of the sphere
9 cm
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37. B) The diameter of a sphere with volume 972 in3
9 cm
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38. C) The volume of the hemisphere
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39. 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.
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40. 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:
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41. Next Page:
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42. 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
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43. Surface Area of a Sphere
The surface area of a sphere with radius r is S = 4 r . 2 r
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44. Finding Surface Area of Spheres
Find each measurement. Give your answers in terms of .
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46. Exploring Effects of Changing Dimensions
The radius of the sphere is tripled. Describe the effect on the volume.
3 m
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47. 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.
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48. 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.
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49. You did a great job today!
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