Lesson 11.1 Meaning of Volume pp. 460-464

Objectives: 1. To state the volume postulates. 2. To apply the Volume Addition Postulate to solids. 3. To prove volume formulas and apply them to various solids.

Definition The volume of a solid is the number of cubic units needed to fill up the interior completely.

Postulate 11.1 Volume Postulate. Every solid has a volume given by a positive real number.

Postulate 11.2 Congruent Solids Postulate. Congruent solids have the same volume.

Postulate 11.3 Volume of Cube Postulate. The volume of a cube is the cube of the length of one edge: V = e3.

Postulate 11.4 Volume Addition Postulate. If the interiors of two solids do not intersect, then the volume of their union is the sum of the volumes.

Theorem 11.1 The volume of a rectangular prism is the product of its length, width, and height: V = lwh.

Find the volume. V = lwh = 4(6)(5) = 120 un.3 5 6 4

Practice: Find the volume. 7 cm 3 cm 11 cm

Practice: Find the volume. 6 5 3 14 8 12 4

Homework pp. 462-464

Find the volume of each solid. 1. ►A. Exercises Find the volume of each solid. 1. 4 V = e3 V = 43 V = 64 un.3

Find the volume of each solid. 3. ►A. Exercises Find the volume of each solid. 3. 5 2 7 V = lwh V = 7(2)(5) V = 70 un.3

►A. Exercises Find the volume of each solid. 5. A cube with edge x. V = e3 V = x3 cu. units

Find the volume of each solid. 9. ►B. Exercises Find the volume of each solid. 9. s 10 s 10 s cube

Find the volume of each solid. 9. s2 + s2 = 102 2s2 = 100 s2 = 50 ►B. Exercises Find the volume of each solid. 9. s s2 + s2 = 102 2s2 = 100 s2 = 50 10 s s = 50 s = 5 2 cube

Find the volume of each solid. 9. V = e3 V = (5 2)3 V = 250 2 ►B. Exercises Find the volume of each solid. 9. 10 s V = e3 V = (5 2)3 V = 250 2 V = 353.553... V ≈ 353.6 un.3 cube

Find the volume of each solid. 10. ►B. Exercises Find the volume of each solid. 10. s2 + s2 = 2s2 s2 + 2s2 = 62 3s2 = 36 6 s s = 12 s s = 2 3 s

►B. Exercises Find the volume of each solid. 11. 32 + w2 = 52 8 3 5 32 + w2 = 52 9 + w2 = 25 w2 = 16 w = 4 w

►B. Exercises Find the volume of each solid. 11. 52 + h2 = 82 3 5 52 + h2 = 82 25 + h2 = 64 h2 = 39 h h = 39 4

►B. Exercises Find the volume of each solid. 11. V = lwh 8 3 5 V = lwh V = (3)(4)( 39) 39 V = 12 39 V ≈ 74.94 un.3 4

►B. Exercises Find the volume of each solid. 13. 3 4 7 7 9 3 V = lwh V = 3(7)(9) V = 189

►B. Exercises Find the volume of each solid. 13. 3 4 7 4 3 V = lwh V = 3(3)(4) V = 36

►B. Exercises Find the volume of each solid. 13. 3 4 7 Vsolid = 189 - 36 Vsolid = 153 un.3

►B. Exercises Find the volume of each solid. 14. 4 5 2 3 3 3

►B. Exercises Find the volume of each solid. 14. 4 2 3 5 2 3 3 5

►B. Exercises Find the volume of each solid. 16. A right rectangular prism has a volume of 3536 cu. feet, and the length of the base of the prism is 9 ft. longer than its width. The height of the prism is 26 ft. What are the dimensions of the base of the prism?

►B. Exercises 16. V = 3536 3536 = lwh 3536 = 26(x+9)(x) x = -17 or x = 8 17 ft.  8 ft.

►C. Exercises 19. s2 +s2 = x2 2s2 = x2 s2 = x2 2 x s s = x 2 2 s

■ Cumulative Review Reread the explanation of the first three area postulates, then explain the following. 21. Which postulates guarantee that areas exist and are meaningful?

■ Cumulative Review Reread the explanation of the first three area postulates, then explain the following. 22. Which postulate provides a first method for finding an area without counting squares?

■ Cumulative Review Reread the explanation of the first three area postulates, then explain the following. 23. Find the area of a regular hexagon with a 16-in. side.

■ Cumulative Review 24. Find the area of the figure in the diagram. 6 8

■ Cumulative Review 25. Find the surface area of the rectangular prism with dimensions l, w, and H.