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Geometry

Geometry. Surface Area of Prisms and Cylinders. Goals. Know what a prism is and be able to find the surface area. Know what a cylinder is and be able to find the surface area. Solve problems using prisms and cylinders. Prism. A polyhedron with two congruent faces, called the bases .

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Geometry

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  1. Geometry Surface Area of Prisms and Cylinders

  2. Goals • Know what a prism is and be able to find the surface area. • Know what a cylinder is and be able to find the surface area. • Solve problems using prisms and cylinders.

  3. Prism • A polyhedron with two congruent faces, called the bases. • The bases are parallel. • The other faces are parallelograms and are called lateral faces. • The segments joining corresponding vertices of the bases are lateral edges.

  4. Example Base Lateral Edges Lateral Face Lateral Face Base

  5. Prisms can have any polygon for its bases. Base is a pentagon. Base is a triangle. Pentagonal Prism Triangular Prism

  6. These are not prisms: Lateral Faces are not parallelograms. …and no parallel bases.

  7. Altitude of a Prism The perpendicular distance between the bases. We usually use the letter h for the height – the length of the altitude. h

  8. Right Prism • The lateral edges are perpendicular to the bases. For clarity, in many cases we do not indicate right prisms – use common sense.

  9. Oblique Prism • A prism in which lateral faces are not perpendicular to the bases. 110

  10. Slant Height Generally, you can use the Pythagorean Theorem to find one or the other. The length of a lateral edge in an oblique prism. Slant Height s Height h

  11. Do you know… • What a prism is? • What the bases are? • What a lateral face is? • What the lateral edges are? • What a right prism is? • What an oblique prism is? • What the slant height is?

  12. Classifying Prisms • Use the shape of the base in the name. Right Triangular Prism Right Rectangular Prism Right Pentagonal Prism

  13. Have you ever seen a regular heptagonal prism?

  14. Surface Area • The sum of the areas of all the faces of a prism. • Area = Area of 2 bases + all lateral faces. • Contrary to the text, use the symbol SA for surface area.

  15. Example 6 The pink sides are really rectangles. They look like parallelograms because of the projection. 4 25 There are 2 bases and 4 lateral faces. All are rectangles.

  16. Example 4 6 4 6 20 25 What’s the area? A = 20  25 = 500 6 6 ? ? 4 4

  17. Example 4 6 4 6 20 Surface Area is the sum of the lateral area (500) and the two bases (48). 25 A = 20  25 = 500 SA = 548 6 6 4 24 4 24

  18. What we did. This measurement is the perimeter of a base. P h A = Ph We found the rectangular area. We found the area of both bases. B B

  19. SA = 2B + Ph The surface area is the sum of these regions. P h A = Ph B B

  20. 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

  21. Alternate Method • Find the area of each face separately. • Add them together. • Don’t omit any face – be careful.

  22. Lateral Area • The lateral area of a shape is the area of the lateral faces, but doesn’t include the bases. • SA = 2B + Ph is total surface area. • Ph is the lateral area. • LA = Ph

  23. Base Example Find the surface area. 2 ft. 2 ft. 12 ft. P = 8 h = 12 B = 4 2B + Ph SA = 2(4) + 8(12) = 8 + 96 SA = 104 ft2 or…

  24. Base Example Find the surface area. Alternate solution. 2 ft. 2 ft. 12 ft. P = 28 h = 2 B = 24 2 B + P h SA = 2(24) + 28(2) = 48 + 56 SA = 104 ft2

  25. Example 2 ft. Alternate solution 2. 2 ft. 12 ft. Separate the figure into a “net”. Find the area of each face. 104 ft2 24 ft2 24 ft2 24 ft2 4 ft2 24 ft2 4 ft2

  26. ExampleFind the Surface Area B = 40 P = 44 h = 16 SA = 2B + Ph SA = 2(40) + 44(16) SA = 80 + 704 S = 784 Base 16 h 2 20

  27. Your Turn Find the surface area. 7 cm 6 cm 18 cm

  28. Solution Perimeter = 2(6 + 18) = 48 cm Area = 6  18 = 108 cm2 Base 7 cm 6 cm 18 cm SA = 2B + Ph = 2(108) + 48(7) = 216 + 336 = 552 cm2 Lateral Area

  29. 6 6 4 6 Find the Surface Area Area of Equilateral Triangular Base B Surface Area Hint:

  30. Try this problem. 12 10 Find the surface area of the right, hexagonal prism. Each base is a regular polygon.

  31. 12 Solution 12 12 ? ? ? 6 10 That’s the area of one base.

  32. Solution SA = 2B + Ph SA = 2(374.1) + 72(10) SA = 748.2 + 720 SA = 1468.3 12 374.1 10 The perimeter of the hexagon is 6  12 = 72, and the height is 10.

  33. Summary • A prism is a polyhedron with 2 congruent bases and parallelogram lateral faces. • Prisms may be right or oblique. • Basic Formula: SA = 2B + Ph • The Lateral Area LA = Ph

  34. Cylinders

  35. Cylinder • A prism with congruent circular bases. • May be right or oblique, just like prisms. r r = radius h = height h

  36. Surface Area of a Cylinder Take a cylinder and cut it apart… You get two circles and a rectangular area.

  37. Surface Area of a Cylinder h 2r The width of the rectangle is… the circumference of the circle.

  38. Surface Area of a Cylinder 2rh h 2r The area of the rectangle is… 2rh (aka Lateral Area)

  39. Surface Area of a Cylinder r 2rh h r2 2r r2 The area of one circle is… r2 The area of two circles is 2r2.

  40. Surface Area of a Cylinder r 2rh h r2 2r r2 The surface area of the cylinder is: SA = 2r2 + 2rh

  41. Surface Area of a Cylinder r Or, for easier computing… h

  42. Example Find the surface area. 12 SA = 2r(r + h) SA = 2(12)(12 + 10) SA = 24(22) SA = 528 SA  1658.76 10

  43. Your Turn Find the surface area. d = 2 in. SA = 2(1)(1 + 14) SA = 2(15) SA = 30 SA  94.25 in2 r = 1 in. 14 in.

  44. Problem. Find the height. 4 h SA = 301.6

  45. 6 h SA = 282.74 Your turn. Find the height.

  46. Take a clean sheet of paper… • Label it Chapter 12 Formulas • Add these formulas: • Prism Cylinder • SA = 2B + Ph SA=2r(r + h) • LA = Ph LA = 2rh • Everyday as you have new formulas, add them to it with a simple drawing.

  47. Practice Problems

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