1. By Kylie Bolton and Jessica Nguyen

2.  Find volumes of spheres and hemispheres.  Solve problems involving volumes of spheres and hemispheres with given diameters, radius’, surface areas, and circumferences.

3.  A sphere is the set of points in space equidistant from a certain point.  It has no faces, edges, or vertices.  An example is our earth.

4. How do we get the equation for the volume of a sphere? Finding the volume of a sphere is kind of like finding the volume of a pyramid and the surface area of a sphere. Suppose that the space inside a sphere is divided into an infinite number of near pyramids with all their vertices on the center of the sphere. The height of the pyramid equals the radius of the sphere. The sum of all the pyramids’ bases will equal the surface area of the entire sphere. The sum of all the pyramids’ volume is equal to the volume of the sphere.

5. Recall, V(pyramid)= 1 / 3 Bh, where B is the area of its base and h is its height. Also, the surface area of a sphere, 4∏r², is equal to B 1 + B 2 + B 3 +…+ B n V= 1 / 3 B ¹ h ¹ + 1 / 3 B ² h ² + 1 / 3 B ³ h ³ +…+ 1 / 3 B n h n = 1 / 3 B 1 r + 1/3 B 2 r + 1/3 B 3 r +…+ 1/3 B n r = 1 / 3 r ( B 1 + B 2 + B 3 +…+ B n ) = 1 / 3 r (4∏ r ²) = 4 / 3 ∏ r ³ Sum of the volumes of all the pyramids. Replace h with r. Distributive Property Replace B 1 + B 2 + B 3 +…+ B n With 4∏ r ² Simplify

6.  If a sphere has a volume of a V cubic units and a radius of r units, then V=4/3 r ³

7.  Find the Volume of the sphere if C=24m  Round to the nearest 10 th Find r C=2∏ r 24=2∏ r R≈3.82m Plug it in! V= 4 / 3 ∏(3.82)³ V=233.5m³

8.  Billy was delivering spherical teapots. He accidently spilt the tea out of one. So he needs to know how much tea to put in the teapots.  Help him out by finding the volume of the spherical teapot if the radius=12in.  Round to the nearest 10 th. 12

9. FFind the volume of the jug. VV=4/312³ VV=7238.2in³   Substitution  Multiply

10.  Hemisphere- half of a sphere.  V=1/2(4/3r³)

11. Example: Find the volume of this hemisphere with a radius of 2in. 2in

12. 1. Plug the radius into the formula V=1/2(4/3r³) 2. V=1/2(4/32³) 3. V=1/2(33.5) 4. V=16.8

13. FIND THE VOLUME OF THE HEMISPHERE IF THE SURFACE AREA=10 ROUND TO THE NEAREST 10TH Example 1:

14.  T=1/2(4 r ²)+  r ²  10=1/2(4 r ²)+  r ²  10=3 r ²  3.333= r ²  1.83= r  V=1/2(4/3 r ³)  V=1/2(4/31.83³)  V=12.8 units²  Surface area of a sphere  Substitute  Divide both sides by 3∏  Find the sq. root  Volume of a sphere  Substitute  Use a calculator

15. Pg. 704 #3-7, 9-20, 23-24, 30-31 By Jessica and Kylie!