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Objectives

7.1 Surface Area and Volume. Objectives. Explore ratios of surface area to volume. Develop the concepts of maximizing volume and minimizing surface area. 7.1 Surface Area and Volume. Theorems, Postulates, & Definitions.

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Objectives

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  1. 7.1 Surface Area and Volume Objectives • Explore ratios of surface area to volume. • Develop the concepts of maximizing volume and minimizing surface area.

  2. 7.1 Surface Area and Volume Theorems, Postulates, & Definitions Surface area and volume of a right rectangular prism 7.1.1: The surface area,S, and volume, V, of a right rectangular prism with length l, width w, and height h are S = 2lw + 2wh + 2lh and V = lwh, respectively. Surface area and volume of a cube 7.1.2: The surface area, S, and volume, V, of a cube with side s are S = 6s2 and V = s3.

  3. 7.1 Surface Area and Volume Key Skills Solve problems by using the ratio of surface area to volume. Find the ratio of surface area to volume for a right rectangular prism with l = 5 meters, w = 8 meters, and h = 10 meters. Surface Area: S = 2lw + 2lh + 2wh S = 2(5)(8) + 2(5)(10) + 2(8)(10) = 340square meters. Volume: V = lwh V = (5)(8)(10) = 400 cubic meters.

  4. Ratio of surface area to volume: 340 17 = = 0.85 400 20 TOC 7.1 Surface Area and Volume Key Skills Solve problems by using the ratio of surface area to volume. Find the ratio of surface area to volume for a right rectangular prism with l = 5 meters, w = 8 meters, and h = 10 meters.

  5. 7.2 Surface Area and Volume of Prisms Theorems, Postulates, & Definitions Volume of a Prism 7.2.3: The volume, V, of a prism with height h and base area B is V = Bh.

  6. Area of each base: 1 (11)(4) = 22 2 7.2 Surface Area and Volume of Prisms Key Skills Find the surface area of a right prism. Surface area: S = hp + 2B. Lateral area: hp or 10(5 + 9 + 11) = 250 Surface area: 250 + 2(22) = 294 square feet.

  7. TOC 7.2 Surface Area and Volume of Prisms Key Skills Find the volume of a prism. Volume: V = Bh Area of the base: 22 square feet Volume: (22)(10) = 220 cubic feet

  8. 1 1 S = L + B or S = lp + B Bh . 2 3 V = . 7.3 Surface Area and Volume of Pyramids Theorems, Postulates, & Definitions Surface Area of a Regular Pyramid 7.3.1: The surface area, S, of a regular pyramid with lateral area L, base area B, perimeterof the base p, and slant height l is given by: Volume of a Pyramid 7.3.2: The volume, V, of a pyramid with height h and base area B is given by:

  9. S = lp + B Surface area: 1 1 1 (15)(72) + 324 = 864 square feet S = lp + B = 2 2 2 7.3 Surface Area and Volume of Pyramids Key Skills Find the surface area of a regular pyramid. Area of base: 182 = 324 square feet Perimeter of base: 4(18) = 72 square feet

  10. 1 1 1 (324)(12) = 1296 cubic feet V = V = Bh Bh = 3 3 3 TOC 7.3 Surface Area and Volume of Pyramids Key Skills Find the volume of a regular pyramid.

  11. 7.4 Surface Area and Volume of Cylinders Theorems, Postulates, & Definitions Surface Area of a Right Cylinder 7.4.1: The surface area, S, of a right cylinder with lateral area L, base area B, radius r, and height h is S = L + 2B or S = 2rh + 2r2. Volume of a Cylinder 7.4.2: The volume, V, of a cylinder with radius r, height h, and base area B is V = Bh or V = r2h.

  12. 7.4 Surface Area and Volume of Cylinders Key Skills Find the surface area of a right cylinder. S = 2rh + 2r2 S = 2rh + 2r2 = 2(4)(10) + 2(42) = 112 351.86 square inches.

  13. TOC 7.4 Surface Area and Volume of Cylinders Key Skills Find the volume of a right cylinder. V = r2h V = r2h =(42)(10) = 160 502.65 cubic inches

  14. 1 1 r2h . V = Bh or V = 3 3 7.5 Surface Area and Volume of Cones Theorems, Postulates, & Definitions Surface Area of a Right Cone 7.5.1: The surface area, S, of a right cone with lateral area L, base area B, radius r, and slant height l is given by: S = L + B or S = rl + r2. Volume of a Cone 7.5.2: The volume, V, of a cone with radius r, height h, and base area B is given by:

  15. Surface area: S = L + B = 65 + 25 = 90  282.74 square meters. 7.5 Surface Area and Volume of Cones Key Skills Find the surface area of a right cone. S = L + B Lateral area: L = rl = (5)(13) = 65 204.20 Area of the base: A = r2 = 25 78.54

  16. V = (5)2(12) = 100 V = 1 1 1 r2h = r2h  314.16 cubic meters 3 3 3 TOC 7.5 Surface Area and Volume of Cones Key Skills Find the volume of a right cone.

  17. 4 r3 . V = 3 . S = 4r2 7.6 Surface Area and Volume of Spheres Theorems, Postulates, & Definitions Volume of a Sphere 7.6.1: The volume, V, of a sphere with radius r is given by: Surface Area of a Sphere 7.6.2: The surface area, S, of a sphere with radius r is given by:

  18. 7.6 Surface Area and Volume of Spheres Key Skills Find the surface area of a sphere. Find the surface area of a sphere with radius 12 feet. S = 4r2 S = 4r2 = 4(12)2 = 576 1809.56 square feet

  19. 4 4 4 (12)3 =2304  7238.23 cubic feet r3 = r3 V = V = 3 3 3 TOC 7.6 Surface Area and Volume of Spheres Key Skills Find the volume of a sphere. Find the volume of a sphere with radius 12 feet.

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