Geometry Finding the Surface Area of Solids
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Surface! Surface!
Finding the Surface Area of Solids

 

 Now we know how to find the volume 

 of some objects.
 
 Next we will see how to find 
 the area of the surface of those objects.
 
 First the cube.
 
 A cube is made up of six equal sized squares ...
 

 
 Find the area of one of the squares
 multiply by six
 and BINGO! 
 You're done.
 
 Example:
 Find the area of the surface of this cube ...
 

 
 The sides are 5 units each.
 The area of one surface is ...

 

  5 5 = 25

 
 There are six faces on the surface
 so the area of the entire surface is ...

25 6 = 150

 
 Example:
 Find the area of the surface
 of this rectangular solid object ...
 

 

 
 There are two ways to calculate this one.
 The top and bottom surfaces are 5 x 5 = 25 units each.
 The four side faces are 8 x 5 = 40 units each.
 

  2 25 + 4 40 = 210

 
 The other way to calculate this one 
 starts out the same way.
 
 The top and bottom surfaces are 5 x 5 = 25 units each.
 Then ...
 The perimeter of the bottom surface
 is 5 + 5 + 5 + 5 = 20 units.
 Multiply this times the height (8)
 and we get the area of the surface
 of all 4 sides all at the same time.
 20 x 8 = 160.
 Add that to the area of the top and bottom ...
 

2 25 + 160 = 210

 
 Example:
 Find the area of the surface of the cylinder
 

 
 This is why we used that goofy second method
 in that last example.
 We need it for stuff like this cylinder.
 
 The areas of the top and the bottom
 of the cylinder are each:
 
  Area = pr2
  Area = 9p
  Area 28.274
 
 To find the area of the side, 
 we first need to find the circumference of the circle ...
 

Circumference = p diameter

 
 The radius equals 3,
 so the diameter equals 6 ...
 
  Circumference = p 6
  Circumference 18.850
 
 The area of the side is equal to
 the circumference of the base
 times the height.
 
  Area of the side 18.850 10
  Area of the side 188.50
 
 The total area of the surface of the cylinder
 is equal to the area of the top and bottom
 plus the area of the side.
 

28.274 + 28.274 + 188.50 = 245.048

 
 Sometimes we just need to find the area on the side.
 Math types wanted a fancy way "the area on the side."
 They came up with the term LATERAL AREA.
 Lateral means something like "on the side."
 
 OK, then give me a ham on rye with LATERAL mayo!
 
 Ha ha ha, how funny. Can we get back to math now?
 
 Just remember that if somewhere (like on a test maybe)
 they ask you for the LATERAL AREA,
 they don't want the are on the top and the bottom.
 They only want the mayo.
 

   copyright 2005 Bruce Kirkpatrick

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