Lesson 11.5 Polyhedra and Spheres pp. 482-487
Objectives: 1. To prove the formula for the volume of a sphere. 2. To find volumes using appropriate formulas.
In this section we will attempt to develop the formula for finding the volume of a sphere.
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.
r t t 2r P r
Concentric circles are circles that have the same center but radii of different lengths. The region bounded by concentric circles is called the annulus.
r t
Aannulus = Alg circle - Asm circle Aannulus = r2 - t2
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.
A B C t x r
Since ABC is a right triangle, the Pythagorean theorem applies. x r
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)
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.
Vsphere = Vcylinder - Vtwo cones
3 1 V = r2(2r) – 2 r2 (r) ÷ ø ö ç è æ 3 2 V = 2r3 - r3 3 2 6 V = r3 - r3 3 4 V = r3
Theorem 11.7 The volume of a sphere is four-thirds times the cube of the radius: 3 4 V = r3
EXAMPLE Find the volume of a sphere with a diameter of 10 inches. V = r3 4 3 V = (5)3 4 3 V ≈ 166.67 cubic inches ≈ 523.6 in.3
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 ≈ 113.1 in.3
Regular Polyhedron Volume V = e3 12 2 tetrahedron cube V = e3 V = e3 3 2 octahedron 3 15 + 7 5 V = ( )e3 dodecahedron 12 15 + 5 5 V = ( )e3 icosahedron
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
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
Homework pp. 485-487
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
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
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 2 units
6. sphere with diameter of 8 3 units ►A. Exercises Give the volume of the sphere or regular polyhedron. 6. sphere with diameter of 8 3 units
►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?
►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)
►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?
►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″
►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?
►B. Exercises 20. 3 cm 10 cm
■ Cumulative Review Identify each term defined below. 24. A line in the plane of a circle that intersects the circle in exactly one point
■ Cumulative Review 25. A triangle with no congruent sides Identify each term defined below. 25. A triangle with no congruent sides
■ Cumulative Review 26. A line that intersects two parallel lines Identify each term defined below. 26. A line that intersects two parallel lines
■ Cumulative Review Identify each term defined below. 27. A region of a circle bounded by a chord and the intercepted arc
■ Cumulative Review Identify each term defined below. 28. A portion of a sphere determined by intersecting great circles