Geometry Surface Area & Volume of Prisms & Cylinders.

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1 Geometry 11.5 Surface Area & Volume of Prisms & Cylinders

2 11.5 Essential Question How do you find the surface area and volume of a prism or cylinder? Geometry 12.2 Surface Area of Prisms and Cylinders 2

3 Goals Know what a prism is and be able to find the surface area and volume. Know what a cylinder is and be able to find the surface area and volume. Solve problems using prisms and cylinders. Geometry 12.2 Surface Area of Prisms and Cylinders 3

4 Prism A polyhedron with two congruent faces, called the bases. The bases are parallel. The segments forming the bases are base edges. The other faces are parallelograms and are called lateral faces. The segments joining corresponding vertices of the bases are lateral edges. Geometry 12.2 Surface Area of Prisms and Cylinders 4

5 Parts of a Prism Base Lateral Face Lateral Edges Lateral Face Base Geometry 12.2 Surface Area of Prisms and Cylinders 5

6 Prism Right Prism - all lateral faces are rectangles. Oblique Prism - has at least one non-rectangular lateral face. Geometry 12.2 Surface Area of Prisms and Cylinders 6

7 How do you find the surface area of a right prism? Geometry 12.2 Surface Area of Prisms and Cylinders 7

8 Prism Surface Area The surface area of a right prism can be found using SA = 2B + PH B is the area of each base P is the perimeter of a base H is the height Geometry 12.2 Surface Area of Prisms and Cylinders 8

9 Volume The number of cubic units contained in a solid. Measured in cubic units. 1 V = 1 cu. unit 1 1 Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 9

10 Prism: V = Bh B = area of the base, h = height B B B h h h Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 10

11 Cavalieri s Principle If the area of cross sections and heights of two solids are equal, then the volumes are equal. Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 11

12 Right Prism SA = 2B + PH V= BH SA = Surface Area P = Perimeter of the Base H = Height of the Prism B = Base Area (Area of the Base) V = volume Geometry 12.2 Surface Area of Prisms and Cylinders 12

13 Example1: Name the solid and then find its surface area and volume Geometry 12.2 Surface Area of Prisms and Cylinders 13

14 Example 2: Name the solid and then find its surface area and volume Geometry 12.2 Surface Area of Prisms and Cylinders 14

15 Example 3: Name the solid and then find its surface area and volume. Triangular Prism Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 15

16 Example 4: Name the solid and then find its surface area and volume ??6 3 Geometry 12.2 Surface Area of Prisms and Cylinders 16

17 Example 5: Name the solid and then find its surface area and volume Geometry 12.2 Surface Area of Prisms and Cylinders 17

18 Example 6 L A metal bar has a volume of 2400 cm 3. The sides of the base measure 4 cm by 5 cm. Determine the length of the bar. 5 4 Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 18

19 Example 6 Solution 4 Method 1 L Method 2 5 V = Bh V = L W H B = 4 5 = = 20h h = 120 cm 2400 = L = 20L L = 120 cm Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 19

20 Cylinder A prism with congruent circular bases. May be right or oblique, just like prisms. Geometry 12.2 Surface Area of Prisms and Cylinders 20

21 Surface Area of a Cylinder Take a cylinder and cut it apart You get two circles and a rectangular area. Geometry 12.2 Surface Area of Prisms and Cylinders 21

22 Surface Area of a Cylinder 2 rh Area of the rectangle. h r 2 r 2 r circumference of the circle. r 2 area of two circle Geometry 12.2 Surface Area of Prisms and Cylinders 22

23 Surface Area of a Cylinder 2 rh h r r 2 2 r The surface area of the cylinder is: SA = 2 r rh r 2 Geometry 12.2 Surface Area of Prisms and Cylinders 23

24 Surface Area of a Cylinder r SA = 2πr 2 + 2πrh h Geometry 12.2 Surface Area of Prisms and Cylinders 24

25 Cylinder: V = r 2 h B r h h V = Bh Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 25

26 Cylinder: SA = 2πr 2 + 2πrH V = πr 2 H r = radius H = Height of the Solid SA = Surface Area V = Volume Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 26

27 Example 7: Name the solid and then find its surface area and volume Geometry 12.2 Surface Area of Prisms and Cylinders 27

28 Example 8: Name the solid and then find its surface area and volume. d = 2 in. r = 1 in. 14 in. Geometry 12.2 Surface Area of Prisms and Cylinders 28

29 Example 9: Find the height. 4 SA = 96π h Geometry 12.2 Surface Area of Prisms and Cylinders 29

30 Example 10 Find the diameter of the can. 3 in V = 115 in 3 Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 30

31 Summary A prism is a polyhedron with 2 congruent bases and parallelogram lateral faces. A cylinder has 2 congruent circular bases, but it is not a polyhedron. Prisms & cylinders may be right or oblique. The volumes of prisms and cylinders are essentially the same: SA = 2B + Ph SA = 2πr 2 + 2πrh Geometry 12.2 Surface Area of Prisms and Cylinders 31

32 Summary The volumes of prisms and cylinders are essentially the same: V = Bh & V = r 2 h where B is the area of the base, h is the height of the prism or cylinder. Use what you already know about area of polygons and circles for B. Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 32

33 Problem 1: A manufacturer of concrete sewer pipe makes a pipe segment that has an outside diameter (o.d.) of 48 inches, an inside diameter (i.d.) of 44 inches, and a length of 52 inches. Determine the volume of concrete needed to make one pipe segment. Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 33

34 Problem 1 Solution Strategy: Find the area of the ring at the top, which is the area of the base, B, and multiply by the height. View of the Base Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 34

35 Problem 1 Solution Strategy: Find the area of the ring at the top, which is the area of the base, B, and multiply by the height. Area of Outer Circle: A out = (24 2 ) = 576 Area of Inner Circle: A in = (22 2 ) = 484 Area of Base (Ring): A Base = = 92 Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 35

36 Problem 1 Solution V = Bh A Base = B = 92 V = (92 )(52) 52 V = 4784 V 15,029.4 in 3 Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 36

37 This one! Problem 2: Which Holds More? 2.3 in 4 in 4.5 in 3.2 in 1.6 in V (3.2)(1.6)(4) V Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 37

38 Problem 3: What would the height of cylinder 2 have to be to have the same volume as cylinder 1? r = 4 r = 3 #1 8 #2 h Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 38

39 Solution r = 4 V # Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 39

40 Solution r = h 128 h #2 h 9 h 14.2 Monday, May 7, 2:54 Geometry 12.4 Volume of Prisms and Cylinders 40

41 Homework Geometry 12.2 Surface Area of Prisms and Cylinders 41

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