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Surface Area & Volume

Surface Area & Volume. Vocabulary. Polyhedron – a three-dimensional figure whose surfaces are polygons Face – each of the polygons of the polyhedron Edge – a segment that is formed by the intersection of two faces Vertex – a point where three or more edges intersect. Vocabulary.

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Surface Area & Volume

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  1. Surface Area & Volume

  2. Vocabulary • Polyhedron – a three-dimensional figure whose surfaces are polygons • Face – each of the polygons of the polyhedron • Edge – a segment that is formed by the intersection of two faces • Vertex – a point where three or more edges intersect

  3. Vocabulary • Polyhedron – a 3-dimensional (3-D) solid with faces and edges

  4. Prisms • A prism is a polyhedron with exactly two congruent, parallel faces, called bases. • Other faces are lateral faces (also congruent) • An altitude of a prism is a perpendicular segment that joins the planes of the bases. • The heighth of the prism is the length of an altitude.

  5. Prisms • Prism – are named by the shape of their bases. • This one is a _________________ prism.

  6. Prisms • Prism – are named by the shape of their bases. • This one is a _________________ prism.

  7. Prisms • Prisms – are named by the shape of their bases. • This one is a _________________ prism.

  8. Prisms • Have bases and faces that are polygons – therefore, these sides have all of the properties of polygons • One of those properties is area.

  9. Prisms • Net – is what we would have if we took a polyhedron an unfolded it. • If we made a cube into a net, here is what we would have:

  10. Prisms • If we made a triangular prism into a net, here is what we would have:

  11. Surface Area of Prisms • Surface area – is just that, the area of the surface of the entire polyhedron. • How many surfaces does this have?

  12. Surface Area of Prisms • How might we go about finding the surface area of this square polyhedron?

  13. Surface Area of Prisms • Surface area = area of all of the lateral faces + area of the two bases

  14. Surface Area of Prisms • The lateral area of a prism is the sum of the areas of the lateral faces. • The surface area is the sum of the lateral area and the area of the two bases.

  15. Surface Area of Prisms • The lateral area of a right prism is the product of the perimeter of the base and the height. LA = ph • The surface area of a right prism is the sum of the lateral area and the areas of the two bases SA = LA + 2B

  16. Surface Areas of Prisms • Use a net to find the surface area. 8 ft 7 ft 6 ft

  17. Surface Areas of Prisms • Use a formula to find the lateral area and surface area. • L.A.=ph=24*7=168 ft2 • S.A.= L.A.+2 B= 216 ft2 8 ft 7 ft 6 ft

  18. Homework • Handout on Surface Area of Prisms

  19. Surface Areas of Cylinders • A cylinder has two congruentparallel bases. However, the bases of a cylinder are a circle. • An altitude of a cylinder is a perpendicular segment that joins the planes of the bases. • The heighth of a cylinder is the length of an altitude.

  20. Surface Areas of Cylinders Right cylinder Oblique cylinder

  21. Surface Areas of Cylinders • The lateral area of a right cylinder is the product of the circumference of the base and the height of the cylinder. LA = 2πrh or LA = πdh • The surface area of a right cylinder is the sum of the lateral area and the areas of the two bases. SA = LA + 2πr2

  22. Surface Areas of Cylinders • The radius of the base of a cylinder is 16 in., and its height is 4 in. Find its surface area in terms of π. S.A.= 2πrh + 2πr2= 640 π

  23. Homework • Handout on surface area of cylinders

  24. Volumes ofPrisms and Cylinders • Volume is the space that a figure occupies. It is measured in cubic units. • Volume of a prism V = Bh B - area of the base For a rectangular prism, V = (l * w) * h

  25. Volumes ofPrisms and Cylinders • Volume of a cylinder V = Bh B - area of the base • In a cylinder, B= πr2, so V = πr2h.

  26. Volumes ofPrisms and Cylinders • Find the volume of the figure to the nearest whole number. V = Bh V= (½ *3*5)*6 V= 45 in.3 6 in. 5 in. 3 in.

  27. Volumes ofPrisms and Cylinders • Find the area of the figure to the nearest whole number. • V = πr2h • V = π(2)25= 63m3 5 m 2 m

  28. Homework • Handouts on the volume of prisms and cylinders

  29. Surface Area • Review: • Prism – a 3-D polyhedron with 2 bases and a height that connects them • Cylinder – a 3-D polyhedron with circles for bases • Surface Area – the area of all of the sides of a polyhedron added together • Net – a polyhedron unfolded into a flat, 2-dimensional figure

  30. Surface Area of Pyramids • A pyramid: • Is a polyhedron • Has only 1 base that is a polygon • Has sides (lateral faces) that are triangles • Has sides that meet at a vertex • Has a height and a slant height • We name pyramids by the shape of the base.

  31. Surface Areas ofPyramids • A regularpyramid is a pyramid whose base is a regular polygon and whose lateral faces are congruent isosceles triangles. • The slant height l is the length of the altitude of a lateral face of the pyramid.

  32. Surface Areas ofPyramids • The lateral area of a regular pyramid is the sum of the areas of the congruent lateral faces. LA = ½ pl p – perimeter of the base l – slant height • The surface area of a regular pyramid is the sum of the lateral area and the area of the base. SA = LA + B

  33. Surface Areas ofPyramids • Find the slant height of a square pyramid with base edges 12 cm and altitude 8 cm. • First, draw and label the figure. • l= 10 cm

  34. Surface Areas ofPyramids • Find the lateral area and surface area of the regular square pyramid below. LA = ½ pl = ½ (4*4 sides) 7 = 56 in.2 SA = LA + B = 56 + (4*4) = 72 in.2

  35. Homework • Handout on surface area of pyramids

  36. Surface Areas ofCones • A cone is like a pyramid, but its base is a circle. • The slant height is the distance from the vertex to a point on the edge of the base.

  37. Surface Areas ofCones • The lateral area of a right cone is half the product of the circumference of the base and the slant height. LA = ½ (2πr)l = πrl • The surface area of a right cone is the sum of the lateral area and the area of the base. SA = LA + πr2

  38. Surface Areas ofCones • Find the lateral and surface area of a cone with radius 8 cm and slant height 17 cm. Leave your answer in terms of π. • LA = πrl • LA= 136π cm2 • SA = LA + πr2 • SA= 136π + 64π= 200πcm2

  39. Surface Area of Cones • Find the Surface Area of a cone with a base with radius 10ft and a slant height of 12ft: • LA = πrl • LA= 120π ft2 • SA = LA + πr2 • SA= 120π+ 100π= 220πft2

  40. Homework • Handout on Surface Area of Cones

  41. Volumes ofPyramids and Cones • Volume of a pyramid V = 1/3 Bh • In a cone, B= πr2, so V = 1/3 πr2h.

  42. Volumes ofPyramids and Cones • Find the volume of the figure. • V = 1/3 Bh • V= 1/3 (6*6) *5 • V= 60 ft3

  43. Volumes ofPyramids and Cones • Find the volume of a cone with diameter 3 m and height 4 m. • V = 1/3 πr2h • V= 1/3 π(3/2)24 • V= 3π m3

  44. Volumes ofPyramids and Cones • Find the volume of a square pyramid with base edges 24 in. long and slant height 15 in. • V = 1/3 Bh • First find h. • 122+h2=152 • h=9 • V= 1/3 (24*24)*9= 1728 in.3

  45. Homework • Handouts on Volume of Pyramids and Cones

  46. Surface Areas and Volumesof Spheres • A sphere is the set of all points in space equidistant from a given point called the center.

  47. Spheres • Half of a sphere is called a hemisphere. • How do you think the volume of a hemisphere would compare to that of a sphere?

  48. Spheres • What is the relationship between a sphere and a cylinder? • Same radius; volume of a sphere is less

  49. Surface Area of a Sphere • A sphere is a 3-Dimensional object and a circle is only 2-D (flat). • What do we think we can say about the surface areas of the two in comparison to each other?

  50. Surface Area of a Sphere • If the surface area of a sphere is more than the area of a circle, then the formula must give us a value greater than that of a circle. • Area of a Circle = π r2 • Surface area of a Sphere = 4(π r2)

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