Surface area and volume of solids. Download the power point presentation to enable animation.
Download as PPTX, PDF1 like1,460 views
The document provides an overview of various three-dimensional geometric figures, specifically focusing on prisms, cylinders, pyramids, and cones. It explains how to calculate their surface areas and volumes using applicable formulas, along with examples for clarity. Detailed exercises are also included to practice finding surface areas and volumes based on given dimensions.
Surface area and volume of solids. Download the power point presentation to enable animation.
- 1. PRISMS
- 2. PRISMS A three dimensional geometric figure with two identical polygonal bases that aligned in parallel and with parallelogram on each side.
- 6. SURFACE AREA OF PRISMS When we want to find the surface area of prisms, we can unfold it so that it becomes many connecting two- dimensional geometric figures and then we calculate the sum of those areas of those figures to obtain the surface area of the original prism.
- 7. unfold Right Triangular Prism c a b h a ba b c c b ba a h 1 2 3 4 5
- 8. a h a a a a a a a a a a h Square Prism 1 2 3 4 5 6
- 9. a b h a b a b a a b a Rectangular Prism 1 2 3 4 5 6
- 11. Examples: 1. Find the surface area of a square prism given that each side of the base is 5 m and the height of the prism is 12 m. 5m 12 m
- 12. Examples: 2. Find the surface area of a right triangular prism that the length of a leg of the right triangle is 5 inches and the length of the other leg is 12 inches while the height of the prism is 16 inches. 12 in 5 in 16 in
- 13. Examples: 3. Find the surface area of a regular hexagonal prism of that the length of each side of the hexagonal base is 2 centimeters and the height is 8 centimeters. 2 cm 8 cm
- 14. 4. Find the surface area of the triangular prism below: Examples:
- 15. Exercises: 1. Find the surface area of a right triangular prism that the length of a leg of the right triangle is 5 inches and the length of the other leg is 12 inches while the height of the prism is 16 inches. 2. Find the surface area of a square prism given that each side of the base is 4 m and the height of the prism is 11 m.
- 16. TRY THIS 1. A house with an equilateral triangular roof and rectangular walls is to be painted with red. What is the surface area of the house if its exterior is to be painted? (Refer to the diagram below) 13 m 9 m 16 m 13 m
- 17. 2. A pipe with a cross-shaped base is meant to connect a water source and a house. What is the exterior surface area of the pipe? (Refer to the diagram below) TRY THIS 4 cm 6 cm 6 cm 2600 cm 4 cm 4 cm 6 cm 6 cm 4 cm
- 18. VOLUME OF PRISMS the amount of space that a substance or object occupies. volumes of prisms can be obtained by multiplying the area of the base to its height.
- 19. Rectangular Prism a b h V=Ah V=abh V is the volume A is the base area h is the height
- 20. Right Triangular Prism a b h V=Ah Rectangular prisms Right Triangular prisms V= 1 2 Ah V= 1 2 abh
- 21. Hexagonal Prism Hexagonal prisms h a Base Area A = 3 3 2 𝑎2 Volume V=Ah V= 3 3 2 𝑎2ℎ
- 22. Examples: 1. Find the volume of a trapezoidal prism of which the length of the parallel lines of the bases are 5 and 8 inches and the height of the trapezoid is 4 inches. The height of the prism is 12 inches. 5 in 8 in 4 in 12 in
- 30. 180
- 32. Stephanie just redecorated her bedroom. She wants to paint her door green. 200 cm 80 cm 3 cm ANSWER: 33,680 cm2
- 33. Jordan built her cat Tuna a new scratching post. She needs to cover the post with carpet. How much carpet does Jordan need to cover the surface of the post, including the bottom? 10 cm 90 cm 10 cm ANSWER: 3,800 cm2
- 34. Nemo's aquarium is filled with 2400 cubic centimeters of water. The base of the aquarium is 20 centimeters long and 12 centimeters wide. What is the height of the water in Nemo’s aquarium? ANSWER: 10 cm 20 cm 12 cm ℎ
- 35. Erin built a rectangular trough for her horses to eat from. The trough is 3 meters long, 2 meters wide, and 2 meters high. Hay for the trough costs $14 per cubic meter. How much will it cost to completely fill the trough with hay? 3 m 2 m 2 m ANSWER: $168
- 36. An exotic fish is placed in a rectangular aquarium that has a length of 75 cm ,width of 35 cm, and a height of 24 cm. If the water level rises 2 cm when the fish is placed, what is the volume of the fish? 75 cm 35 cm 2 cm 24 cm ANSWER: 5,250 cu. ft
- 37. A shipping company wants to ship its boxes in a 8-feet wide, 20-feet long, and 9-feet high trailer truck. The boxes have dimensions of 2 feet by 5 feet by 3 feet (2ft × 5ft × 3ft). What is the total volume of boxes that can fit in the trailer if they have to be stacked so that the bottom of each box measures 3 feet wide and 5 feet long and two boxes are placed in each row? 8 ft 20 ft 5 ft 3 ft 9 ft 2 ft ANSWER: 960 cu. ft
- 38. Cylinder Cross section or base axis height Right cylinder is a three-dimensional geometric figure with parallel cross sections or bases. Each bases of the right cylinder is vertical to its axis. Oblique cylinder is a three-dimensional geometric figure with parallel cross sections or bases. Each bases of the oblique cylinder is, however, not vertical to its axis.
- 39. Surface Areas of Cylinders We can find the surface area of a cylinder by unfolding the cylinder into two-dimensional geometric figures and find the total surface areas of the figures. h r
- 40. Example 1 Find the surface area of a cylinder whose base diameter is 14 centimeters and the height is 4 centimeters. 𝜋 ≈ 22 7 4 cm 14 cm
- 41. Example 2 Find the surface area of a three-dimensional geometric figure that each base of this figure is half of a 3.5 centimeter radius circle and the length of it is 12 centimeters long as shown in the figure below. 𝜋 ≈ 3.5 cm 12 cm
- 42. Example 3 A cylindrical trash can is 4 feet high with 8 feet diameter. If the side, not the top nor the bottom, of the can has to be painted with the paint that costs 30 baht per square foot, how much money is needed? 𝜋 ≈ 22 7 8 ft 4 ft
- 43. Find the outer surface area 𝜋 ≈ 22 7
- 44. Find the inner surface area 𝜋 ≈ 3.14 interior
- 45. Find the inner surface area 𝜋 = 22 7 interior
- 46. Volume of Cylinders pentagon hexagon octagon nonagon decagon
- 47. A polygon with an infinite number of sides is called a circle.
- 48. Volume of Prisms Area of the base heightV=Ah A h Volume of Cylinders V=Ah A h = 𝜋𝑟2 V=𝜋𝑟2 h
- 49. Example 1 Find the volume of the cylinder whose diameter is 4 centimeters and height is 9 centimeters. 𝜋 ≈ 22 7 9cm 4 cm
- 50. Example 2 Find the height of a cylinder whose diameter is 8 inches and volume is 120 cubic inches. 𝜋 ≈ 22 7 h 8 inches
- 51. Example 3 Find the amount of plastic material used in making a water pipe with the outer diameter of 2 centimeters, the thickness of 2 millimeters, and the length of 25 centimeters. 𝜋 ≈ 22 7 25 cm 2 cm 1 cm
- 52. 1. A company is deciding which box to use for their merchandise. The first box measures 8 inches by 6 inches by 10 inches. The second box measures 9 inches by 5 inches by 12 inches. If each box costs 25 baht per square inch to make, how much does the company save by choosing the cheaper box if the company needs fifty boxes?
- 53. 2. A cosmetics company that makes small cylindrical bars of soap wraps the bars in plastic prior to shipping. Find the surface area of a bar of soap if the diameter is 14 centimeters and the height is 12 cm. 𝜋 ≈ 22 7
- 54. 3. A farmer needs to make a barn using the plan below. How much wood does he need if he has to cover the exterior of the barn? 13 13 10 18 16
- 55. A shipping company wants to ship its boxes in a 8-feet wide, 20-feet long, and 9-feet high trailer truck. The boxes have dimensions of 2 feet by 5 feet by 3 feet (2ft × 5ft × 3ft). What is the total volume of boxes that can fit in the trailer if they have to be stacked so that the bottom of each box measures 3 feet wide and 5 feet long and two boxes are placed in each row? 8 ft 20 ft 5 ft 3 ft 9 ft 2 ft ANSWER: 960 cu. ft
- 56. 5. A cylindrical golden tube with a diameter of 21 inches and a height of 9 inches is to be melted into right triangular golden prims, wherein the legs of its triangular base are 8 inches and 15 inches and its lateral area is 240 square inches. How many right triangular golden prisms can be made from the cylindrical golden tube?
- 57. PYRAMI D
- 58. Pre-test A square pyramid has a base with the side length of 6 centimeters and the slant height of 5 centimeters. Find the surface area of the pyramid.
- 61. Apex Lateral Edge Lateral Face Height Base Right Pyramid Oblique Pyramid
- 63. Surface Area PYRAMID S of To find the surface area of a pyramid, we can unfold it and find the sum of the areas from two- dimensional figures.
- 64. a a 𝑙 a 𝑙 Slant height Equilateral Triangular Pyramid
- 65. a 𝑙 Slant height Square Pyramid 𝑙 a a
- 67. Example 1 A square pyramid has a base with the side length of 6 centimeters and the slant height of 5 centimeters. Find the surface area of the pyramid.
- 68. 5 cm 6 cm 5 cm 6 cm Example 1 A square pyramid has a base with the side length of 6 centimeters and the slant height of 5 centimeters. Find the surface area of the pyramid.
- 69. Example 2 The base of an equilateral triangular pyramid has the side length of 8 centimeters. All of the lateral faces of the pyramid, including the base, are identical. Find the surface area of the pyramid.
- 70. Example 2 The base of an equilateral triangular pyramid has the side length of 8 centimeters. All of the lateral faces of the pyramid, including the base, are identical. Find the surface area of the pyramid. 8 cm 8 cm 8 cm 8 cm
- 71. Example 3 A rectangular pyramid whose base is 4 centimeters wide and 6 centimeters long and has the length of the lateral edges of 8 centimeters. Find the surface area of the pyramid.
- 72. Example 3 A rectangular pyramid whose base is 4 centimeters wide and 6 centimeters long and has the length of the lateral edges of 8 centimeters. Find the surface area of the pyramid. 6 cm 6 cm 4 cm 4 cm 8 cm 8 cm 𝑙1 𝑙2 8 cm
- 74. Volume of Pyramids 1 3 V = Ah_ base area height
- 75. Examples: 1. Find the volume of a 10 centimeteres high reactangular pyramid whose base is 6 centimeters wide and 8 centimeters long. 10 cm 8 cm 6cm
- 76. Examples: 2. A pyramid with a square base is 6 centimeters high and has the volume of 32 cubic centimeters. Find the a) base area of the pyramid. b) the length of each side of the base. 6 cm x x V = 32 cu. cm
- 80. CONES
- 81. Cones A three-dimensional geometric figure with a circular base. Apex Slant height Axis Base Height Height Right Cone Oblique Cone A right cone has its height which is vertical to the base through the center of the base. Its height and its axis are the same line. All line segments connecting the apex to any points on the perimeter of the base are equal in length and called slant heights. An oblique cone has its height which is also vertical to the base but not at the center of the base. Its height is not the same line as the axis
- 82. Surface Areas of Cones r h 𝑙 r 2𝜋r 𝑙 2𝜋r
- 83. _____________ __ 2𝜋𝑟 𝑙 C L Area of L__________ Area of C = Arc length of L Perimeter of C _____________ __ Area of L__________ 𝜋𝑙2 = 2𝜋𝑟 2𝜋𝑙 Area of L = 2𝜋𝑟 𝜋𝑙2 2𝜋𝑙 _____________ __ Area of L = 𝜋𝑟𝑙 Surface Areas of Cones
- 84. Area of L = 𝜋𝑟𝑙 𝑙 2𝜋r r 2𝜋r Area of B = Surface Areas of Cones 𝜋𝑟2
- 85. Surface Areas of Cones r h 𝑙 Surface Area = Area of B +Area of L = 𝜋𝑟2 + 𝜋𝑟𝑙
- 86. Example 1 Find the surface area of a cone whose base radius is 7 centimeters and the slant height is 15 centimeters. 𝜋 ≈ 22 7
- 87. Volume of Cones
- 88. Volume of Cones 1 3 V = Ah_ base area height 1 3 V = 𝜋r2 h_
- 89. Example 2 Find the volume of a cone whose base radius is 3.5 centimeters and the height is 12 centimeters. 𝜋 ≈ 22 7
- 90. Surface Areas of Spheres