1. Solids Surface Area and Volume Chapter 12 D. Prescott BEHS

2. Polyhedron Definition: A 3-D region bounded by polygons

3. Classification All polyhedra are either convex or nonconvex (concave) Convex polyhedron – any 2 of its surface points can be connected by a segment that lies entirely inside or on the polyhedron Nonconvex

4. Regular polyhedron All of its polygonal surfaces are: congruent to each other and equilateral and equiangular

5. Platonic Solids are regular polyhedra

6. All “sides” are called faces. They are always in the shape of a polygon. An edge is a segment formed by the intersection of 2 faces. Terms: A vertex is a point where 3 or more edges meet.

7. Euler’s Theorem The number of faces (F) plus the number of vertices (V) is always equal to two more than the number of edges (E). F + V = E + 2 5 + 6 = 9 + 2 11 = 11

8. Cross Section Definition – the intersection of a plane and a solid

9. Click Below for Prisms or Pyramids Prism Presentation Pyramid Tutorial

10. Prisms Measurement of Prisms Including Surface Area and Volume

11. Table of Contents (Prisms) I. Prisms Definition Basic Terms for Prisms –Faces, Edges, HeightFacesEdgesHeight Classification of Prisms –Base Shape, Right, ObliqueBase ShapeRightOblique Surface Area of Prisms –Definition, Lateral AreaDefinitionLateral Area –Total Area, ExampleTotal AreaExample Volume of Prisms –DefinitionDefinition –Formula, ExampleFormulaExample

12. Prism – a polyhedron that is composed of 2 congruent parallel faces and parallelograms for the remaining faces.

13. In every prism there are always 2 faces that are congruent polygons lying in parallel planes. These faces are called bases. In this figure the 2 bases are trapezoids located in the front and back. BASES

14. Names of Prisms Prisms are named by the shape of their bases. Octagonal prism Trapezoidal prism Rectangular prism

15.  FACES  In any prism the lateral faces are parallelograms. In any right prism the lateral faces are rectangles.

16. EDGES The intersection of any 2 faces is a segment that is called an edge. If it is a side of a base, then it is called a base edge. The intersection of 2 lateral faces is called a lateral edge.

17. Height of a Prism The altitude of a prism is a segment joining the 2 bases and is perpendicular to both. The height of a prism is called an altitude.

18. If the lateral faces are rectangles, then the height will be congruent to a lateral edge. Any prism that has rectangular lateral faces is called a right prism. Right Prisms

19. If the lateral faces are not rectangles, then the altitude will not be congruent to a lateral edge. Any prism that does not have rectangular lateral faces is called an oblique prism. Oblique Prisms

20. Classify the Prisms Right Pentagonal PrismRight Rectangular Prism

21. Surface Area of Prisms The surface area of a prism can be found by finding the area of each face and then adding them together. However, this technique can be quite time consuming. This octagonal prism has 8 lateral faces and 2 bases for a total of 10 faces. You would have to find the area of 10 polygons and then add the 10 numbers together.

22. Formula for Lateral Area h e3e3 e4e4 e2e2 e1e1 Face 1: e 1 h Face 2: e 2 h Face 3: e 3 h Face 4: e 4 h Lateral Area = e 1 h + e 2 h + e 3 h + e 4 h LA = (e 1 + e 2 + e 3 + e 4 )h LA = p h

23. Total Surface Area h The total surface area(TSA) is equal to the lateral area(LA) plus the area of the Base(B) times 2. The Bases are congruent polygons. TSA = LA + 2B Or TSA = pH + 2B e3e3 e4e4 e2e2 e1e1 p=perimeter of the base H=Height of the prism B= Area of the base

24. Find the Total Surface Area 7cm 3cm 5cm TSA = 20cm 5cm + 2(21cm 2 ) p = 7+3+7+3 3 7 7 3 TSA = 100cm 2 + 42cm 2 TSA = 142cm 2 TSA = pH + 2B =H B = 21cm 2 p = 20cm B = 7cm 3cm

25. Find the surface area of the triangular prism: 6cm 8cm20cm TSA = pH + 2B =H

26. First find the perimeter and area of the base: 6cm 8cm p = a + b + c c 2 = a 2 + b 2 c 2 = (6) 2 + (8) 2 c 2 = 36 + 64 c 2 = 100 c = 10 p = 6 + 8 + 10 p = 24cm

27. p=24cm, B=24cm 2 and H 6cm 8cm20cm TSA = pH + 2B =20cm S = (24cm)(20cm) + 2(24cm 2 ) TSA = 480cm 2 + 48cm 2 TSA = 528cm 2

28. Surface Area of Cylinders Proof of formula based on prisms formula: TSA = pH + 2B (prisms)

29. A net diagram is a 2-dimensional sketch made by making imaginary cuts along some of the solid’s edges and “unfolding” it. Net Diagrams can be used to find the surface area of a 3-D figure:

30. h A rea = l ength · w idth 2πrh h r Total Surface Area of Cylinders TSA=2πrh + 2πr 2 Area = 2πr·h

31. Find the Total Surface Area 10cm 12cm

32. Pyramids Measurement of Pyramids including Volume and Surface Area by Donna B. Prescott

33. Table of Contents (Pyramids) II. Pyramids Historical Connections –Mayan PyramidsMayan Pyramids –Egyptian PyramidsEgyptian Pyramids Geometrical Terms –Faces, Vertex, and Regular PyramidsFaces, Vertex, and Regular Pyramids –ClassificationClassification –Slant Height and AltitudeSlant Height and Altitude Right Triangles in Regular Pyramids –Formed with the AltitudeFormed with the Altitude –Formed with the Slant HeightFormed with the Slant Height Surface Area –Lateral AreaLateral Area –Total AreaTotal Area Volume Credits References

34. Mexico’s Pyramids The ancient Mayans also built pyramids. It is believed that their pyramids had celestial significance (GeoCities.com).

35. Egyptian Pyramids “The ancient Egyptians built pyramids as tombs for the pharaohs and their queens”(British Museum, 1999).

36. Geometrical Terms All pyramids have triangular lateral faces which intersect in a single point called the vertex of the pyramid. Pyramids are classified according to the shape of the base. Regular pyramids have regular polygons for their bases. Vertex

37. Classify the following pyramids. Triangular pyramid Hexagonal pyramid

38. Altitude and Slant Height V B A The length of the segment joining the vertex (V) and the center of the base (A) is the altitude of the pyramid. It is perpendicular to the base in a right pyramid. The length of the segment from the vertex (V) to the midpoint of a base edge (B) is called slant height.

39. Right Triangles in Regular Pyramids A B V Given: A regular square pyramid with base edges 8cm and altitude 12cm How long is the slant height? 8 12 4 l l=slant height

40. Another Right Triangle Find the length of a lateral edge. 8 E B V 4 c V EB 4

41. Formula for Surface Area TSA = LA + B V LA = ½ e 1 l + ½ e 2 l + ½ e 3 l + ½ e 4 l LA = ½(e 1 + e 2 + e 3 + e 4 )l LA = ½ pl TSA = ½ pl + B l=slant height p =perimeter of the base B = area of the Base

42. Find the Area TSA= ½ pl + B B = 8(8) B = 64cm 2 p = 4(8) p = 32cm TSA= ½·(32)( ) + 64 TSA=(64 + 64)cm 2 V l = 8cm

43. Terminology in Cones 6 in 20 in Cones also have a slant height, the cone’s height, and a vertex. Vertex h Right Triangles in Cones: 20 in h 6 in Slant height Cone’s height l

44. Surface Area of a Cone TSA= ½ pl + B (pyramids) TSA= ½ cl + B TSA= ½ (2πr)l + πr 2 TSA= (½ ·2)πrl + πr 2 TSA= πrl + πr 2

45. Find the surface area: 6 in 20 in TSA= πrl + πr 2 TSA= π(6)(20) + π(6) 2 TSA = 120π + 36π TSA = 156π in 2

46. VOLUME OF SOLIDS Section 12.4 Volume of Prisms and Cylinders Page 743 Mrs. Prescott, BEHS

47. Volume of Prisms h e3e3 e4e4 e2e2 e1e1 Volume of a prism is measured in cubic units, and measures the space inside of the figure. A cubic unit is like an ice cube which measures 1 unit in length, width, and height. Finding the volume would be like counting the number of ice cubes in a stack of ice trays. 1 1 1

48. Volume Formula h e3e3 e4e4 e2e2 e1e1 The number of cubes that would take to fill up the bottom row of the figure can be determined by finding the area of the base(B). The number of rows is equal to the height of the prism. Volume is equal to the area of the prism’s base(B) times the prism’s height(H). V= BH

49. Find the Volume of the Prism 7cm 5cm V= BH 3cm 7cm Base = 21cm 2 Height = 5cm V = 21cm 2 5cm V=105cm 3 3cm

50. Find the Volume: 5cm 10cm18cm V= BH = height

51. Volume of A Cylinder V= BH Base is a Circle B = _________

52. Find the Volume 10cm 14cm

53. Volume of a Pyramid The formula for finding the volume of a pyramid is one-third times the area of the base times the pyramid’s altitude. A pyramid can hold one- third the volume of a prism with the same base area and height.

54. Given: A regular square pyramid with base edges 8cm and altitude 12cm H = 12cm B = 64cm 2 8cm 12cm

55. Volume of a Cone h r

56. Find the Cone’s Volume 6 in 20 in First find the cone’s height 20in h 6 in 20in

57. 6 in 20 in

58. Spheres Definition –The set of all points that are a given distance from a set point

59. Interesting Facts The Planet Earth, our home, is nearly a sphere, except that it is squashed a little at the poles. It is a spheroid, which means it just misses out on being a sphere because it isn't perfect in one direction (in the Earth's case: North- South)

60. Largest Volume for Smallest Surface Put another way it can contain the greatest volume for a fixed surface area If you blow up a balloon it naturally forms a sphere because it is trying to hold as much air as possible with as small a surface as possible. The sphere appears in nature whenever a surface wants to be as small as possible. Examples include bubbles and water drops.

61. Terms in spheres: Center Radius Diameter Chord B A O C O E D

62. Great Circle & Hemisphere In geometry a hemisphere is an exact half of a sphere. It also refers to half of the Earth, such as the "Northern Hemisphere” A Great Circle is a cross section that passes through the sphere’s center and divides the sphere into 2 congruent halves.

63. Formulas for Spheres How are they derived?

64. Formulas for Spheres

65. r h

66. The total surface area of a sphere: The volume of a sphere:

67. Given: r = 10cm, find the sphere’s surface area and volume.

69. Given: the sphere’s surface area is 800π m 2, find its radius. 4π4π 4π4π