Surface Area of a Rectangular SolidDate: 09/21/1999 at 19:34:32 From: Sarah Schuldenfrei Subject: Surface area of solids Hi, My math teacher told us that we could calculate the surface area of a cube or a 3-dimensional rectangular object by calculating the area of the sides you can see and multiplying by 2. She said this didn't work for more complicated shapes. But I thought about it and I think it will work for any 3-dimensional object that you can build using cubes. Is this right? Sarah Date: 09/30/1999 at 13:25:20 From: Doctor TWE Subject: Re: Surface area of solids Hi Sarah. Thanks for writing to Dr. Math. You have to be careful about how you're looking at the rectangular solid when applying your teacher's "shortcut." If you're looking directly at the center of one face, the shortcut won't work, because you only see that one face, like this: +-----+ | | | | +-----+ If you're looking along the centerline of one face (but not at the center of the face), you'll only see two faces, like this: +-----+ | | +-----+ | | | | +-----+ Only if you're looking at the rectangular solid "off center" will you see 3 of the 6 faces, like this: +-----+ / /| +-----+ | | | + | |/ +-----+ In this case, you teacher's shortcut will work. The shortcut works due to symmetry. The three faces you don't see match one-to-one with the three faces you do see. This won't necessarily work with other 3D figures, however, because they aren't necessarily symmetrical. Consider taking a cube made up of a block of 3x3x3 smaller unit cubes, and removing one edge cube from the hidden side, as diagrammed below. Each of the three faces you see has a surface area of 3x3 = 9 square units, for a total surface area of 3*9 = 27 square units. One of the hidden faces remains a 3x3 square (surface area = 9 square units), but the other two have been altered. There are 8 unit squares along each of their flat faces, but where the other square was "removed," there are now four unit squares in the indented hollow spot. They are the three marked with an "x" in the diagram, plus one opposite the "9" on the lower left corner cube (marked 9,7,7). The total surface area of the figure is 27+9+8+8+4 = 56, which is more than 2*27 = 54. Front Back (hidden) +-----+-----+-----+ +-----+-----+-----+ / 1 / 2 / 3 /| /| 1 | 2 | 3 | +-----+-----+-----+3| +3| | | | / 4 / 5 / 6 /| + /| +-----+-----+-----+ +-----+-----+-----+2|/| +2|/| 4 | 5 | 6 | / 7 / 8 / 9 /| +6| /| +6| | | | +-----+-----+-----+1|/| + +1|/| +-----+-----+-----+ | 1 | 2 | 3 | +5|/| | +5|/| 7 | x /| 8 | | | | |/| +9| |/| +9| |---+x| | +-----+-----+-----+4|/| + +4|/| +-----+ | +-----+ | 4 | 5 | 6 | +8|/ | +8|/ 7 / x |/ 8 / | | | |/| + |/| +-----+-----+-----+ +-----+-----+-----+7|/ +7|/ 4 / 5 / 6 / | 7 | 8 | 9 | + | +-----+-----+-----+ | | | |/ |/ 1 / 2 / 3 / +-----+-----+-----+ +-----+-----+-----+ I hope this helps! - Doctor TWE, The Math Forum http://mathforum.org/dr.math/ |
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