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  1. Assignment • P. 806-9: 2-20 even, 21, 24, 25, 28, 30 • P. 814-7: 2, 3-21 odd, 22-25, 30 • Challenge Problems: 3-5, 8, 9

  2. In Glorious 3-D! Most of the figures you have worked with so far have been confined to a plane—two-dimensional. Solid figures in the “real world” have 3 dimensions: length, width, and height.

  3. Polyhedron A solid formed by polygons that enclose a single region of space is called a polyhedron. Separate your Geosolids into 2 groups: Polyhedra and others.

  4. Parts of Polyhedrons • Polygonal region = face • Intersection of 2 faces = edge • Intersection of 3+ edges = vertex edge face vertex

  5. Warm-Up Separate your Geosolidpolyhedrainto two groups where each of the groups have similar characteristics. What are the names of these groups? Prisms Pyramids Polyhedra:

  6. 12.2 & 12.3: Surface Area of Prisms, Cylinders, Pyramids, and Cones Objectives: • To find and use formulas for the lateral and total surface area of prisms, cylinders, pyramids, and cones

  7. Prism A polyhedron is a prismiff it has two congruent parallel bases and its lateral faces are parallelograms.

  8. Classification of Prisms Prisms are classified by their bases.

  9. Right & Oblique Prisms Prisms can be right or oblique. What differentiates the two?

  10. Right & Oblique Prisms In a right prism, the lateral edges are perpendicular to the base.

  11. Pyramid A polyhedron is a pyramid iff it has one base and its lateral faces are triangles with a common vertex.

  12. Classification of Pyramids Pyramids are also classified by their bases.

  13. Pyramid A regular pyramid is one whose base is a regular polygon.

  14. Pyramid A regular pyramid is one whose base is a regular polygon. • The slant height is the height of one of the congruent lateral faces.

  15. Solids of Revolution The three-dimensional figure formed by spinning a two dimensional figure around an axis is called a solid of revolution.

  16. Cylinder A cylinder is a 3-D figure with two congruent and parallel circular bases. • Radius = radius of base

  17. Cone A cone is a 3-D figure with one circular base and a vertex not on the same plane as the base. • Altitude = perpendicular segment connecting vertex to the plane containing the base (length = height)

  18. Cone A cone is a 3-D figure with one circular base and a vertex not on the same plane as the base. • Slant height= segment connecting vertex to the circular edge of the base

  19. Right vs. Oblique What is the difference between a right and an oblique cone?

  20. Right vs. Oblique In a right cone, the segment connecting the vertex to the center of the base is perpendicular to the base.

  21. Nets Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid. An unfolded pizza box is a net!

  22. Nets Imagine cutting a 3-D solid along its edges and laying flat all of its surfaces. This 2-D figure is a net for that 3-D solid.

  23. Activity: Red, Rubbery Nets Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net?

  24. Activity: Red, Rubbery Nets Match one of the red, rubbery nets with its corresponding 3-D solid. Which of the shapes has no net? Asphere doesn’t have a true net; it can only be approximated.

  25. Exercise 1 There are generally two types of measurements associated with 3-D solids: surface area and volume. Which of these can be easily found using a shape’s net?

  26. Surface Area The surface area of a 3-D figure is the sum of the areas of all the faces or surfaces that enclose the solid. • Asking how much surface area a figure has is like asking how much wrapping paper it takes to cover it.

  27. Lateral Surface Area The lateral surface area of a 3-D figure is the sum of the areas of all the lateral faces of the solid. • Think of the lateral surface area as the size of a label that you could put on the figure.

  28. Exercise 2 What solid corresponds to the net below? How could you find the lateral and total surface area?

  29. Exercise 3 Draw a net for the rectangular prism below. A B D C To find the lateral surface area, you could: • Add up the areas of the lateral rectangles

  30. Exercise 3 Draw a net for the rectangular prism below. Height of Prism Perimeter of the Base To find the lateral surface area, you could: • Find the area of the lateral surface as one, big rectangle

  31. Exercise 3 Draw a net for the rectangular prism below. Height of Prism Perimeter of the Base To find the total surface area, you could: • Find the lateral surface area then add the two bases

  32. Surface Area of a Prism Lateral Surface Area of a Prism: • P = perimeter of the base • h = height of the prism Total Surface Area of a Prism: • B = area of the base

  33. Exercise 4 Find the lateral and total surface area.

  34. Exercise 5 Draw a net for the cylinder. Notice that the lateral surface of a cylinder is also a rectangle. Its height is the height of the cylinder, and the base is the circumference of the base.

  35. Exercise 6 Write formulas for the lateral and total surface area of a cylinder.

  36. Surface Area of a Cylinder Lateral Surface Area of a Cylinder: • C = circumference of base • r = radius of base • h = height of the cylinder Total Surface Area of a Cylinder:

  37. Exercise 7 The net can be folded to form a cylinder. What is the approximate lateral and total surface area of the cylinder?

  38. Height vs. Slant Height By convention, h represents height and l represents slant height.

  39. Height vs. Slant Height By convention, h represents height and l represents slant height.

  40. Exercise 8 Draw a net for the square pyramid below. To find the lateral surface area: • Find the area of one triangle, then multiply by 4

  41. Exercise 8 Draw a net for the square pyramid below. To find the lateral surface area: • Find the area of one triangle, then multiply by 4

  42. Exercise 8 Draw a net for the square pyramid below. To find the total surface area: • Just add the area of the base to the lateral area

  43. Surface Area of a Pyramid Lateral Surface Area of a Pyramid: • P = perimeter of the base • l = slant height of the pyramid Total Surface Area of a Prism: • B = area of the base

  44. Exercise 9 Find the lateral and total surface area.

  45. Exercise 10 You may have realized that the formula for the lateral area for a prism and a cylinder are basically the same. The same is true for the formulas for a pyramid and a cone. Derive a formula for the lateral area of a cone. Lateral area of a Pyramid: Lateral area of a Cone:

  46. Surface Area of a Cone Lateral Surface Area of a Cone: • r = radius of the base • l = slant height of the cone Total Surface Area of a Cone:

  47. Exercise 11 A traffic cone can be approximated by a right cone with radius 5.7 inches and height 18 inches. To the nearest tenth of a square inch, find the approximate lateral area of the traffic cone.

  48. Tons of Formulas? Really there’s just two formulas, one for prisms/cylinders and one for pyramids/cones.

  49. Assignment • P. 806-9: 2-20 even, 21, 24, 25, 28, 30 • P. 814-7: 2, 3-21 odd, 22-25, 30 • Challenge Problems: 3-5, 8, 9