1. Lesson 11.5
Polyhedra and Spheres pp

2. Objectives: 1. To prove the formula for the volume of a sphere.
2. To find volumes using appropriate formulas.

3. In this section we will attempt to develop the formula for finding the volume of a sphere.

4. Postulate 11.5
Cavalieri’s Principle. For any two solids, if all planes parallel to a fixed plane form sections having equal area, then the solids have the same volume.

5. r t t 2r P r

6. Concentric circles are circles that have the same center but radii of different lengths. The region bounded by concentric circles is called the annulus.

7. r t

8. Aannulus = Alg circle - Asm circle
Aannulus = r2 - t2

9. Now consider a sphere with radius r and a secant plane passing through it at a distance of t units from the center. The secant intersects the sphere to form circle C with radius x.

10. A B C t x r

11. Since ABC is a right triangle, the Pythagorean theorem applies.x r

12. Since ABC is a right triangle, the Pythagorean theorem applies.
t2 + x2 = r2
Solving for x2
x2 = r2 - t2
Since the area of this sector is a circle
Asection = x2
Substituting for x2
Asection = r2 - t2
Asection = (r2 - t2)

13. x t t t r 2r r r r
The volume of a sphere is equal to the volume of the solid between the cones and the cylinder.

14. Vsphere = Vcylinder - Vtwo cones

15. 3 1
V = r2(2r) – r2 (r) ÷ ø ö ç è æ 3 2
V = 2r r3 3 2 6
V = r r3 3 4
V = r3

16. Theorem 11.7
The volume of a sphere is four-thirds  times the cube of the radius: 3 4
V = r3

17. EXAMPLE Find the volume of a sphere with a diameter of 10 inches.
V = r3 4 3
V = (5)3 4 3
V ≈  cubic inches ≈ in.3

18. Practice: Find the volume of a sphere with a diameter of 6 inches.
V = r3 4 3
V = (3)3 4 3
V = 36 cubic inches ≈ in.3

19. Regular Polyhedron Volume
V = e3 12 2
tetrahedron
cube
V = e3
V = e3 3 2
octahedron 3
V = ( )e3
dodecahedron 12
V = ( )e3
icosahedron

20. Formulas for Area
Square A = s2
Rect. & Parallelogram A = bh
Triangle A = ½bh
Trapezoid A = ½h(b1 + b2)
Rhombus A = ½d1d2
Regular Polygon A = ½ap
Circle A = r2
Equilateral Triangle A = s2 4 3

21. 2. rectangular prism V = lwh 3. prism V = BH 4. cylinder V = r2H
Formulas for Volume
1. cube V = e3
2. rectangular prism V = lwh
3. prism V = BH
4. cylinder V = r2H
5. pyramid V = BH
6. cone V = r2H
7. sphere V = r3 1 3 4

22. Homework pp

23. 1. sphere with a radius of 18 feet
►A. Exercises
Give the volume of the sphere or regular polyhedron.
1. sphere with a radius of 18 feet

24. 2. sphere with a radius of meter 1 4
►A. Exercises
Give the volume of the sphere or regular polyhedron.
2. sphere with a radius of meter 1 4

25. 5. octahedron with an edge of 2 units
►A. Exercises
Give the volume of the sphere or regular polyhedron.
5. octahedron with an edge of units

26. 6. sphere with diameter of 8 3 units
►A. Exercises
Give the volume of the sphere or regular polyhedron.
6. sphere with diameter of units

27. ►A. Exercises
Give the volume of the sphere or regular polyhedron.
11. A volleyball has a circumference of 27 inches. How many cubic inches of air are needed to inflate the ball?

28. ►B. Exercises
17. A spherical water tower has a diameter of 75 feet. How many gallons of water will it hold? (1 gallon = cubic feet)

29. ►B. Exercises
18. A ball whose diameter is 8 inches is placed in a cube whose edge measures 8 inches. How many cubic inches of sand will fill the box containing the ball?

30. ►B. Exercises
19. A metal part is made in the shape of a cylinder with a hemisphere (half of a sphere) on top. Find the volume of the part. 8″ 4″

31. ►B. Exercises
20. An ice-cream cone looks like the following diagram. Approximately how many cubic centimeters of ice cream are used to fill an ice-cream cone like this one?

32. ►B. Exercises
20.
3 cm
10 cm

33. ■ Cumulative Review
Identify each term defined below.
24. A line in the plane of a circle that intersects the circle in exactly one point

34. ■ Cumulative Review 25. A triangle with no congruent sides
Identify each term defined below.
25. A triangle with no congruent sides

35. ■ Cumulative Review 26. A line that intersects two parallel lines
Identify each term defined below.
26. A line that intersects two parallel lines

36. ■ Cumulative Review
Identify each term defined below.
27. A region of a circle bounded by a chord and the intercepted arc

37. ■ Cumulative Review
Identify each term defined below.
28. A portion of a sphere determined by intersecting great circles