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Volume of Polyhedra and Spheres

This lesson focuses on developing the formula for finding the volume of a sphere and includes examples and practice problems for finding the volumes of spheres and regular polyhedra.

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Volume of Polyhedra and Spheres

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  1. Lesson 11.5 Polyhedra and Spheres pp. 482-487

  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. C x B t r A

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

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

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

  17. V = r3 4 3 4 3 V = (5)3 10 EXAMPLE Find the volume of a sphere with a diameter of 10 inches. V ≈ 166.67 cubic inches ≈ 523.6 in.3

  18. V = r3 4 3 4 3 6 V = (3)3 Practice: Find the volume of a sphere with a diameter of 6 inches. V = 36 cubic inches ≈ 113.1 in.3

  19. 2 2 V = e3 12 V = e3 3 15 + 7 5 V = ( )e3 3 15 + 5 5 V = ( )e3 12 Regular Polyhedron Volume tetrahedron cube V = e3 octahedron dodecahedron 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 3 4

  21. 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 1 3 4 3

  22. Homework pp. 485-487

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

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

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

  26. ►A. Exercises Give the volume of the sphere or regular polyhedron. 6. sphere with diameter of 8 3 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 = 0.13398 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. 4″ 8″ ►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.

  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. 3 cm 10 cm ►B. Exercises 20.

  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 Identify each term defined below. 25. A triangle with no congruent sides

  35. ■ Cumulative Review 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

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