1. Chapter 12 Surface Area and Volume

2. Chapter 12 Objectives Define polyhedron Define polyhedron Utilize Euler’s Theorem Utilize Euler’s Theorem Identify cross sections Identify cross sections Identify a prism and cylinder Identify a prism and cylinder Identify a pyramid and cone Identify a pyramid and cone Calculate the surface area of special polyhedra Calculate the surface area of special polyhedra Calculate the volume of special polyhedra Calculate the volume of special polyhedra Identify a sphere Identify a sphere Calculate the surface area and volume of a sphere Calculate the surface area and volume of a sphere Compare quantities between similar polyhedra Compare quantities between similar polyhedra

3. Lesson 12.1 Exploring Solids

4. Lesson 12.1 Objectives Define polyhedron Define polyhedron Use properties of polyhedra Use properties of polyhedra Utilize Euler’s Theorem Utilize Euler’s Theorem

5. Polyhedron A polyhedron is a solid made of polygons. A polyhedron is a solid made of polygons. Remember, polygons are 2-D shapes with line segments for sides. Remember, polygons are 2-D shapes with line segments for sides. The polygons form faces, or sides of the solid. The polygons form faces, or sides of the solid. An edge of a polyhedron is the line segment that is formed by the intersection of 2 faces. An edge of a polyhedron is the line segment that is formed by the intersection of 2 faces. Typically the sides of the polygon faces. Typically the sides of the polygon faces. A vertex of a polyhedron is a point in which 3 or more edges meet A vertex of a polyhedron is a point in which 3 or more edges meet Typically the corners of the polygon faces. Typically the corners of the polygon faces.

6. Example of a Polyhedron Notice that a polyhedron is nothing more than a 3-dimensional shape. It has some form of a height, a width, and a depth. The plural of polyhedron is polyhedra. Faces Edges Vertices

7. Example 1 Determine if the following figures are polyhedra. Explain your reasoning. No There are no faces. There are no polygons. Yes All faces are polygons. No One of the faces is not a polygon.

8. Regular vs Convex A polyhedron is regular if ALL of its faces are regular polygons. A polyhedron is regular if ALL of its faces are regular polygons. Some polyhedron are regular, some are not! Some polyhedron are regular, some are not! A polyhedron is convex if no line segment connecting two interior points leaves the polyhedron and re- enters. No dents! If there are dents, then the polyhedron is concave.

9. Cross Sections When you take a plane and cut through a solid, the resulting shape of the surface is called the cross section. When you take a plane and cut through a solid, the resulting shape of the surface is called the cross section. When asked to identify a cross section, you need to identify the polygon formed. When asked to identify a cross section, you need to identify the polygon formed. The plane acts like a knife blade and cuts through the solid. The plane acts like a knife blade and cuts through the solid.

10. Theorem 12.1: Euler’s Theorem The number of faces (F), the number of vertices (V), and the number of edges (E) in a polyhedron are related by The number of faces (F), the number of vertices (V), and the number of edges (E) in a polyhedron are related by F + V = E + 2 F + V = E + 2 This is commonly used to find one of the variables above. This is commonly used to find one of the variables above. Typically this is because it is hard to see all the faces, vertices, and edges. Typically this is because it is hard to see all the faces, vertices, and edges. F = 6 V = 8 E = ??? 6 + 8 = E + 2 14 = E + 2 12 = E

11. Example 2 Find the number of vertices, faces, and edges each polyhedron has. F + V = E + 2 F + 8 = 12 + 2 F + 8 = 14 F = 6 F = 5 V = 6 5 + 6 = E + 2 11 = E + 2 E = 9 F = 6 V = 6 6 + 6 = E + 2 12 = E + 2 E = 10 F = 8 V = 12 8 + 12 = E + 2 20 = E + 2 E = 18 You do not have to use the formula to find edges every time. You can use it to find any of the missing quantities as long as you know the other 2, if you use the formula at all!

12. Regular Polyhedra There are only five regular polyhedra, called Platonic solids. There are only five regular polyhedra, called Platonic solids. Named after Greek mathematician and philosopher Plato Named after Greek mathematician and philosopher Plato They use regular triangles, squares, and regular pentagons to form solids. They use regular triangles, squares, and regular pentagons to form solids. Those are the only 3 shapes that can be used. Those are the only 3 shapes that can be used. That is because when their vertices are butted together, their interior angles can add to 360 o. That is because when their vertices are butted together, their interior angles can add to 360 o.

13. Platonic Solids Name Face Shape #Faces#Vertices#Edges TetrahedronTriangle446 CubeSquare6812 OctahedronTriangle8612 DodecahedronPentagon122030 IcosahedronTriangle201230

14. Homework 12.1 In Class In Class 1-9 1-9 p723-726 p723-726 HW HW HW 10-55, 60-70 ev 10-55, 60-70 ev Due Tomorrow Due Tomorrow

15. Lesson 12.2 Surface Area of Prisms and Cylinders

16. Lesson 12.2 Objectives Identify a prism Identify a prism Calculate the surface area of a prism Calculate the surface area of a prism Identify a cylinder Identify a cylinder Calculate the surface area of a cylinder Calculate the surface area of a cylinder Construct a two-dimensional net for three-dimensional solids Construct a two-dimensional net for three-dimensional solids

17. Prisms A prism is a polyhedron with two congruent faces that are parallel to each other. A prism is a polyhedron with two congruent faces that are parallel to each other. The congruent faces are called bases. The congruent faces are called bases. The bases must be parallel to each other. The bases must be parallel to each other. The other faces are called lateral faces. The other faces are called lateral faces. These are always rectangles or parallelograms or squares. These are always rectangles or parallelograms or squares. When naming a prism, they are always named by the shape of their bases. When naming a prism, they are always named by the shape of their bases. Triangular Prism

18. Parts of a Prism The edges that connect opposing bases are called lateral edges. The edges that connect opposing bases are called lateral edges. The height of a prism is the perpendicular distance between the bases. The height of a prism is the perpendicular distance between the bases. In a right prism, the length of the lateral edge is the height. In a right prism, the length of the lateral edge is the height. A right prism is one that stands up straight with the lateral edges perpendicular to the bases. A right prism is one that stands up straight with the lateral edges perpendicular to the bases. In an oblique prism, the height must be drawn in so that it is perpendicular to both bases. In an oblique prism, the height must be drawn in so that it is perpendicular to both bases. An oblique prism is one that is slanted to one side or the other. An oblique prism is one that is slanted to one side or the other. The length of the slanted lateral edge is called the slant height. The length of the slanted lateral edge is called the slant height.

19. Examples of Prisms slant height lateral edge base lateral face

20. Example 3 Answer the following questions: a) What kind of figure is the base? a) Hexagon b) What kind of figure is each lateral face? b) Rectangle c) How many lateral faces does the figure have? c) 6 i. Same number as the sides of the base! d) Give the mathematical name of the geometrical solid. d) Hexagonal Prism

21. Area The surface area of a prism is the sum of the areas of all the faces and bases. The surface area of a prism is the sum of the areas of all the faces and bases. The lateral area of a prism is the sum of the areas of the lateral faces. The lateral area of a prism is the sum of the areas of the lateral faces.

22. Theorem 12.2: Surface Area of a Prism The surface area (S) of a right prism can be found using the formula The surface area (S) of a right prism can be found using the formula S = 2B + Ph S = 2B + Ph B is area of the base B is area of the base A = ½ aP (any polygon) A = ½ aP (any polygon) A = lw (rectangle) A = lw (rectangle) A = bh (parallelogram) A = bh (parallelogram) A = ½ bh (triangle) A = ½ bh (triangle) A = s 2 (square) A = s 2 (square) P is perimeter of the base P is perimeter of the base h is height of the prism h is height of the prism

23. Example 4 Find the surface area of the following: S = 2B + Ph S = 2(56) + Ph S = 2(56) + (5+6+5+6)h S = 2(56) + (5+6+5+6)(7) S = 2(30) + (22)(7) S = 60 + 154 S = 214 m 2 area S = 2B + Ph S = 2( 1 / 2125) + Ph S = 2(56) + (5+12+ )h S = 2(56) + (30)h S = 2(30) + (30)(10) S = 60 + 300 S = 360 m 2 ? a 2 + b 2 = c 2 5 2 + 12 2 = c 2 25 + 144 = c 2 169 = c 2 c = 13 13

24. Cylinder A cylinder is a solid with congruent and parallel circles for bases. A cylinder is a solid with congruent and parallel circles for bases. height

25. Theorem 12.3: Surface Area of a Cylinder The surface area (S) of a right cylinder is The surface area (S) of a right cylinder is S = 2B + Ch S = 2B + Ch B is area of the base B is area of the base A =  r 2 A =  r 2 C is circumference of the base C is circumference of the base C = 2  r or C = 2  r or C =  d C =  d h is height of the cylinder h is height of the cylinder

26. Example 5 Find the surface area of the following cylinder. Round your answer to the nearest hundreth. S = 2B + Ch S = 2  r 2 + Ch S = 2  r 2 + 2  rh S = 2  (9) 2 + 2  (9)(7) S = 2  (81) + 2  (63) S = 162  + 126  S = 288  S = 508.94 + 395.84 = 904.78 ft 2

27. Nets A net is a two-dimensional drawing of a three- dimensional solid. A net is a two-dimensional drawing of a three- dimensional solid. If you were to unfold a solid, the net would show what it looks like. If you were to unfold a solid, the net would show what it looks like. Every solid has a net. Every solid has a net. However, there are only certain ways to draw a net for each solid. However, there are only certain ways to draw a net for each solid.

28. Example 6 Identify the solid formed by the given net. Remember: Solids have a full name (2 parts) Triangular PrismSquare Prism a.k.a Cube Cylinder

29. Homework 12.2 In Class In Class 1,3-12 1,3-12 p730-734 p730-734 HW HW HW 13-37, 43, 44, 50-60 ev 13-37, 43, 44, 50-60 ev Due Tomorrow Due Tomorrow

30. Lesson 12.3 Surface Area of Pyramids and Cones

31. Lesson 12.3 Objectives Identify a pyramid Identify a pyramid Calculate slant height Calculate slant height Calculate the surface area of a pyramid Calculate the surface area of a pyramid Identify a cone Identify a cone Calculate the surface area of a cone Calculate the surface area of a cone

32. Pyramid A pyramid is a polyhedron with one base and lateral faces that meet at one common vertex. A pyramid is a polyhedron with one base and lateral faces that meet at one common vertex. The base must be a polygon. The base must be a polygon. Not necessarily a square! Not necessarily a square! The lateral faces will always be triangles. The lateral faces will always be triangles. Name the pyramid by its base shape. Name the pyramid by its base shape. Square Pyramid Pentagonal Pyramid

33. Regular Pyramids A regular pyramid has a regular polygon for a base, and the common vertex is directly above the center of the base. A regular pyramid has a regular polygon for a base, and the common vertex is directly above the center of the base. The height of a pyramid is the perpendicular distance from the base to the common vertex. The height of a pyramid is the perpendicular distance from the base to the common vertex. The height of a regular pyramid is the length of the line drawn from the center of the base straight up to the common vertex. The height of a regular pyramid is the length of the line drawn from the center of the base straight up to the common vertex. The slant height only exists in regular pyramids and is the length of a line drawn from the base up the lateral face to the common vertex. The slant height only exists in regular pyramids and is the length of a line drawn from the base up the lateral face to the common vertex. Base Slant height Height Lateral edge

34. Finding Slant Height To find the slant height, you must know or be able to calculate To find the slant height, you must know or be able to calculate height of the pyramid height of the pyramid apothem of the base of the pyramid apothem of the base of the pyramid can be found knowing one side of the base. can be found knowing one side of the base. The reason you need those quantities is because they form a hidden right triangle. The reason you need those quantities is because they form a hidden right triangle. Then Pythagorean Theorem can be used to find the missing slant height. Then Pythagorean Theorem can be used to find the missing slant height. Slant height Height Apothem

35. Example 7 Find the slant height for the following pyramid. Remember, you must make a right triangle using height, slant height, and apothem. 10 24 x c 2 = a 2 + b 2 x 2 = 10 2 + 24 2 x 2 = 100 + 576 x 2 = 676 x = √676 x = 26 cm

36. Theorem 12.4: Surface Area of a Regular Pyramid The surface area (S) of a regular pyramid is The surface area (S) of a regular pyramid is S = B + ½P l S = B + ½P l B is the base B is the base P is the perimeter of the base P is the perimeter of the base l is the slant height l is the slant height

37. Example 8 Find the surface area of the following pyramid. S = B + ½P l S = (99) + ½P l S = (99) + ½(9+9+9+9) l S = (99) + ½(9+9+9+9) (10) S = (81) + ½(36) (10) S = (81) + ½(360) S = (81) + 180 S = 261 m 2 area

38. Cone A cone has a circular base and a vertex that is not in the same plane as the base. A cone has a circular base and a vertex that is not in the same plane as the base. All the same rules apply with a cone as they do with a pyramid. All the same rules apply with a cone as they do with a pyramid. height B B

39. Theorem 12.5: Surface Area of a Cone The surface area of a right cone is The surface area of a right cone is S =  r 2 +  r l S =  r 2 +  r l r is radius of the base circle r is radius of the base circle l is the slant height l is the slant height

40. Homework 12.3 In Class In Class 1-13 1-13 p738–741 p738–741 HW HW HW 14-39,44-46,50-53 14-39,44-46,50-53 Due Tomorrow Due Tomorrow

41. Lesson 12.4 Volume of Prisms and Cylinders

42. Lesson 12.4 Objectives Utilize the Volume Postulates Utilize the Volume Postulates Calculate the volume of a prism Calculate the volume of a prism Calculate the volume of a cylinder Calculate the volume of a cylinder Apply Cavalieri’s Principle Apply Cavalieri’s Principle

43. Volume The volume of any solid is the amount of space contained in its interior. The volume of any solid is the amount of space contained in its interior. The volume is measured in cubed units The volume is measured in cubed units m 3 m 3 cm 3 cm 3 ft 3 ft 3 in 3 in 3 units 3 units 3

44. Volume Postulates Postulate 27: Volume of a Cube The volume of a cube is the cube of the length of its side. The volume of a cube is the cube of the length of its side. V = s 3 V = s 3 4 in V = (4 in) 3 V = 64 in 3

45. Volume Postulates Postulate 28: Volume Congruence Postulate 28: Volume Congruence If two polyhedra are congruent, then they have the same volume. If two polyhedra are congruent, then they have the same volume. Postulate 29: Volume Addition The volume of a solid is the sum of the volume of all its nonoverlapping parts.

46. Theorem 12.6: Cavalieri’s Principle If two solids have the same height and the same cross-sectional area at every level, then they have the same volume. If two solids have the same height and the same cross-sectional area at every level, then they have the same volume. So whether the solid is tilted or straight up, the volume is the same as long as the base area is the same size all the way up the solid. So whether the solid is tilted or straight up, the volume is the same as long as the base area is the same size all the way up the solid. B B B height

47. Volume Theorems Theorem 12.7: Volume of a Prism Theorem 12.7: Volume of a Prism The volume (V)of a prism is The volume (V)of a prism is V = Bh V = Bh V is volume V is volume B is area of the base B is area of the base h is the height of the prism h is the height of the prism Theorem 12.8: Volume of a Cylinder The volume (V) of a cylinder is V = Bh B is area of the circular base  r 2 B height B

48. Example 9 Find the volume of the following figures. V = Bh V = (  r 2 )h V = (  (7) 2 )h V = (  (7) 2 )(2) V = (49  )(2) V = 98  = 307.88 cm 3 V = Bh V = []h V = [ 1 / 2 a(ns)]h V = []h V = [ 1 / 2 (2.8)(ns)]h V = []h V = [ 1 / 2 (2.8)(5)(s)]h V = []h V = [ 1 / 2 (2.8)(5)(4)]h V = [](6) V = [ 1 / 2 (2.8)(5)(4)](6) V = [](6) V = [28](6) V = 168 m 3

49. Homework 12.4 In Class 1-9 p746-749 H H WWWW 10-37, 48, 49, 52-60 ev Due Tomorrow

50. Lesson 12.5 Volume of Pyramids and Cones

51. Lesson 12.5 Objectives Calculate the volume of a pyramid Calculate the volume of a pyramid Calculate the volume of a cone Calculate the volume of a cone

52. The Ghost of Cavalieri Cavalieri’s Principle is stated for ALL solids, which include pyramids and cones Cavalieri’s Principle is stated for ALL solids, which include pyramids and cones So the volume of a pyramid or cone is the same as long as the area of the base is congruent and the solids have the same height So the volume of a pyramid or cone is the same as long as the area of the base is congruent and the solids have the same height It does not matter if the solid is slanted or not, it only matters with overall height and area of the base. It does not matter if the solid is slanted or not, it only matters with overall height and area of the base.

53. Volume Theorems Theorem 12.9: Volume of a Pyramid Theorem 12.9: Volume of a Pyramid The volume (V) of a pyramid is The volume (V) of a pyramid is V = 1 / 3 Bh V = 1 / 3 Bh V is volume V is volume B is area of the base B is area of the base h is height of the pyramid h is height of the pyramid Theorem 12.10: Volume of a Cone The volume (V) of a cone is V = 1 / 3 Bh B is area of the circular base  r 2 height B B

54. Example 10 Find the volume of the following solids. V = 1 / 3 Bh V = 1 / 3 [ 1 / 2 bh]h V = 1 / 3 [ 1 / 2 (4)h]h V = 1 / 3 [ 1 / 2 (4)(6)]h V = 1 / 3 [ 1 / 2 (4)(6)](5) V = 1 / 3 [12](5) V = 1 / 3 (60) = 20 m 3 V = 1 / 3 Bh V = 1 / 3 [  r 2 ]h V = 1 / 3 [  (3) 2 ]h V = 1 / 3 [  (3) 2 ](8) V = 1 / 3 [9  ](8) V = 1 / 3 (72  ) V = 24  = 75.40 mm 3

55. Homework 12.5 In Class In Class 1-7 1-7 p755-758 p755-758 HW HW HW 8-25, 30-34, 40-50 ev 8-25, 30-34, 40-50 ev Due Tomorrow Due Tomorrow

56. Lesson 12.6 Surface Area and Volume of Spheres

57. Lesson 12.6 Objectives Define a sphere Define a sphere Calculate the surface area of a sphere Calculate the surface area of a sphere Calculate the volume of a sphere Calculate the volume of a sphere

58. Sphere A sphere is a set of points in space that are equidistant from one given point. A sphere is a set of points in space that are equidistant from one given point. A sphere is a shell of points that are the same distance from the center. A sphere is a shell of points that are the same distance from the center. A sphere is a 3-dimensional circle. A sphere is a 3-dimensional circle.

59. Parts of a Sphere The point inside the sphere where all points are equidistant to is called the center of the sphere. The point inside the sphere where all points are equidistant to is called the center of the sphere. A radius of the sphere is a segment drawn from the center to a point on the sphere. A radius of the sphere is a segment drawn from the center to a point on the sphere. A chord of a sphere is a segment that joins any two points on the sphere. A chord of a sphere is a segment that joins any two points on the sphere. The diameter is also a chord. The diameter is also a chord. Any 2-dimensional circle that contains the center of the sphere is called a great circle. Any 2-dimensional circle that contains the center of the sphere is called a great circle. The equator would be a great circle. The equator would be a great circle. Every great circle of a sphere splits a sphere into two congruent halves called hemispheres. Every great circle of a sphere splits a sphere into two congruent halves called hemispheres.

60. Example 11 Identify the following characteristics: a) Name the center of the sphere. a) T b) Name a segment that is the radius of the sphere. b) segment TS segment TQ segment TP c) Name a chord of the sphere. c) segment QR segment PS d) Find the circumference of the great circle. Write your final answers in terms of . d) 14  m

61. Theorem 12.11: Surface Area of a Sphere The surface area (S) of a sphere is The surface area (S) of a sphere is S = 4  r 2 S = 4  r 2

62. Theorem 12.12: Volume of a Sphere The volume (V) of a sphere is The volume (V) of a sphere is V = 4 / 3  r 3 V = 4 / 3  r 3

63. Example 12 Find the surface area and volume of the following sphere. Round answer to the nearest hundredth. S = 4  r 2 S = 4  (19) 2 S = 4  (361) S = 1444  S = 4536.46 ft 2 V = 4 / 3  r 3 V = 4 / 3  (19) 3 V = 4 / 3  (6859) V = ( 27,436 / 3 )  V = 28,730.91 ft 3

64. Homework 12.6 In Class In Class 1-9 1-9 p762-765 p762-765 HW HW HW 10-28 ev, 45-48, 54-57 10-28 ev, 45-48, 54-57 Due Tomorrow Due Tomorrow

65. Lesson 12.7 Similar Solids

66. Lesson 12.7 Objectives Identify similar solids Identify similar solids Find and use the scale factor of similar solids Find and use the scale factor of similar solids

67. Similar Solids Two solids with equal ratios of corresponding linear measures are said to be similar solids. Two solids with equal ratios of corresponding linear measures are said to be similar solids. That means that the two similar solids must have the same scale factor ratio for: That means that the two similar solids must have the same scale factor ratio for: corresponding heights corresponding heights corresponding radii corresponding radii corresponding side lengths corresponding side lengths

68. Example Determine if the solids are similar. If so, find the scale factor. Not similar Because not all scale factors are equal 1:3 1:4 Similar All spheres are similar! Similar Scale factor is 3:2 [--4--] Similar Scale factor is 5:2

69. Theorem 12:13 Similar Solids Theorem If two solids have a scale factor of a:b, then the corresponding areas have a ratio of: If two solids have a scale factor of a:b, then the corresponding areas have a ratio of: a 2 : b 2 a 2 : b 2 And the corresponding volumes have a ratio of: And the corresponding volumes have a ratio of: a 3 : b 3 a 3 : b 3

70. Homework 12.7 In Class In Class 1-8 1-8 p769-772 p769-772 HW HW HW 9-36, 40-48 ev 9-36, 40-48 ev Due Tomorrow Due Tomorrow