1. Volume of Solid Figures Section 3.8 Standard: MCC9-12.G.GMD.1-3 Essential Questions: How do I derive use the volume formulas, including Cavalieri’s Principle, to find the volume of cylinders, pyramids, cones, and spheres?

2. 3.8A VOLUME OF A SPHERE

3. A sphere is formed by revolving a circle about its diameter. In space, the set of all points that are a given distance from a given point, called the center. Definition: Formula: Volume of a Sphere Spheres

4. 1. Find the volume of the sphere, given that the radius is 8 inches. V ≈ 2144.66 in. 3 8

5. 2. Find the volume of the sphere, given that the diameter is 10 inches. V ≈ 523.60 in. 3 d = 2r 10 = 2r 5 = r 10

6. 3. Find the volume of the sphere, given that the circumference of the sphere is ft. C = 2  r 8  = 2  r 4 = r V ≈ 268.08 ft 3

7. Complete the table. Sphere 1Sphere 2Ratio of each for Sphere 1: Sphere 2 Radius 232 : 3 Diameter Circumference Area of Great Circle Surface Area Volume 464:6 or 2:3 44 66 4  :6  or 2:3 44 99 4  :9  or 4:9 16  36  16  :36  or 4:9

8. Complete the table. Sphere 1Sphere 2Ratio of each for Sphere 1: Sphere 2 Radius aba : b Diameter Circumference Area of Great Circle Surface Area Volume 2a2a2b2ba : b 2a2a2b2b  a2 a2  b2 b2 a 2 : b 2 4a24a2 4b24b2 a 3 : b 3

9. Scale Factor _____________ Area Ratios _______________ Volume Ratios _______________ a : b a 2 : b 2 a 3 : b 3

10. 4. Two spheres have diameters 24 and 36. a. What is the ratio of the areas? b. What is the ratio of the volumes? Ratio of volumes: Radii: Ratio of radii: Ratio of areas: 24 and 36 24 : 36 or 2 : 3 2 2 : 3 2 or 4 : 9 2 3 : 3 3 or 8 : 27

11. 5. A sphere has a radius of 6 meters. The radius of a second sphere is 3 meters. (a.) How does the surface area of the second sphere compare the to surface area of the first sphere? (b.) Volumes? Ratio of radii: 3 : 6 (a). Ratio of areas: 1 2 : 2 2 or 1 : 2 or 1 : 4 The second sphere is ¼ the size of the first. (The area of the second sphere is 4 times smaller) (b). Ratio of volumes: 1 3 : 2 3 or 1 : 8 The second sphere is 1/8 the size of the first. (The volume of second sphere is 8 times smaller)

12. 6. The radius of a sphere is 2.4 cm. (a.) How will the surface area change if the radius is doubled? (b.) Volume? Double the radius: 2(2.4) = 4.8 Ratio of radii:2.4 : 4.8 or 1 : 2 Ratio of surface areas: 1 2 : 2 2 or 1 : 4 Ratio of volumes:1 3 : 2 3 or 1 : 8 The surface area is 4 times larger if the radius is doubled. The volume is 8 times larger if the radius is doubled.

13. 7. Find the surface area of one hemisphere of a circle if the circumference of a great circle of the sphere is 7  cm. C = 2  r 7  = 2  r 7/2 = r Surface area of entire sphere: S = 4  r 2 S = 4  (3.5) 2 S = 49  ½S ≈ 76.93 cm 2 Area of base of hemisphere: A =  r 2 A =  (3.5) 2 A = 12.25  A ≈ 38.48 cm 2 Surface area of hemisphere ≈ 76.93 cm 2 + 38.48 cm 2 ≈ 115.41 cm 2

14. 3.8B VOLUME OF CYLINDERS, CONES, PRISMS, & PYRAMIDS

15. Cylinders are right prisms with circular bases. Therefore, the formula for Volume can be used for cylinders. Volume (V) = Bh = Formula: V = Cylinders h 2πr h

16. Example 8 For the cylinder shown, find the volume. 4 cm 3 cm V = πr 2 h V = π(3) 2 (4) V = 36π

17. Cones Cones are right pyramids with a circular base. Volume (V) = The base area is the area of the circle: Notice that the height (h) (altitude), the radius and the slant height create a right triangle. Formulas: V = l r h

18. 6 cm Example 9: For the cone shown, find the volume. V= 96π cubic cm. 8

19. Volume of a Right Prism (V )= Bh (h = height of prism, B = base area) hPhP hBhB Triangular Right Prism PRISMS Prism: A solid with Bases which are parallel and congruent polygons and Lateral faces which are parallelograms.

20. Example 10: 6 8 5 4 4 B = ½ (6)(4) = 12 V = 12 x 4 = 48 cubic units h = 4 V= Bh

21. STICKY NOTE PROBLEM Find each volume to the nearest tenth. Use 3.14 for . 1. cylinder: radius = 6 m, height = 11 m 1,243.4 m 3 1,114.6 cm 3 612 ft 3 Course 2 904.3 m 3 2. rectangular prism: length = 10 cm, width = 8.64 cm, height = 12.9 cm 3. triangular prism: base area = 34 ft 2, height = 18 ft 4. cylinder: diameter = 8 m, height = 18 m Sticky Note Problem

22. definition: the surface of a conic solid whose base is a polygon. Lateral side vertex altitude Slant height Base PYRAMIDS

23. Pyramid ( B = base area) The volume of a pyramid (V)= ⅓ Bh Volume = ⅓ ⅓⅓ ⅓ (100)(12) == 400 cubic units 12 10 13 Example 11:

24. 6 10 9 m l 53 Sticky Note Problem

25. 3.8C Cavalieri’s Principle

26. A cross section (of a geometric solid): the intersection of a plane and the solid. A prism has the same cross section (parallel to the base) all along its length ! Shown here are the cross sections (in the same plane) of two prisms of equal height. The cross section slices are indicated in red (grey in notes) and are parallel to the bases. If the areas of these two cross section slices are equal, the prisms will be equal in volume.

27. Cavalieri's Principle: If, in two solids of equal height, the cross sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal. Seventeenth century mathematician, Bonaventura Cavalieri, generalized this concept for solids. A generalized statement of this principle: Two prisms will have equal volumes if their bases have equal area and their altitudes (heights) are equal.

28. Use Cavalieri’s Principle to find the volume of the oblique prism. Volume = (6X4)(3) == 72 mm3 V= Bh