Surface Area of Rectangular Prism is sum of areas of 6 rectangles that form the prism

[openyourminds]

I am studying in a GED book. They have explained how to get the surface area of a rectangular prism. I don't understand their way of doing it, and was wondering if you could help me understand. I looked up online and found an alternative way, which makes sense to me. Here are the two different methods.

GED book says:

"The surface area of a rectangular prism is the sum of the areas of the six rectangles that form the prism.

Example 2: Find the surface area of this box.

SA = ph + 2B
SA = perimeter x height + 2(area of base)
SA = (5m + 6m + 5m + 6m) x 7m + 2(6m x 5m)
SA = (22m)7m + 2(30m^2) = 154m^2 + 60m^2 = 214m^2"



Online it says:

Surface Area of a Rectangular Prism = 2ab + 2bc + 2ac

(a, b, and c are the lengths of the 3 sides)
In words, the surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same.
The area of the top and bottom (side lengths a and c) = a*c. Since there are two of them, you get 2ac. The front and back have side lengths of b and c. The area of one of them is b*c, and there are two of them, so the surface area of those two is 2bc. The left and right side have side lengths of a and b, so the surface area of one of them is a*b. Again, there are two of them, so their combined surface area is 2ab.



Thanks!
Nicole

 

[stapel]

I am studying in a GED book. They have explained how to get the surface area of a rectangular prism. I don't understand their way of doing it, and was wondering if you could help me understand. I looked up online and found an alternative way, which makes sense to me. Here are the two different methods.

GED book says:

"The surface area of a rectangular prism is the sum of the areas of the six rectangles that form the prism.

Example 2: Find the surface area of this box.

SA = ph + 2B
SA = perimeter x height + 2(area of base)...

Well, that's certainly not how I would have introduced the topic, no. :shock:

First, they're assuming that you were given the perimeter value, rather than the values of the length and the width, which of course makes no sense, because you then wouldn't have any way to find the area of the bases. Second, they're jumping right to a shortcut (their favorite?), which leaves the reasoning hidden.

In effect, they've taken the prism and taken off the top and the bottom. These have the same area, so they've found the area of one of them, and then multiplied by 2. Then they took the rest, the "sides", if you will, cut along one of the edges, and unfolded. Laying this flat, you then have a length equal to what had been the perimeter (the distance around) and width equal to what had been the height. They found this area. Then they added the two values.

You should use whatever method you best understand, and always remember (especially in case you forget a formula) that the surface area of anything covered in polygons (like rectangles) can be found by finding the area of each face, and then adding.

 

[openyourminds]

Hmm

Well, that's certainly not how I would have introduced the topic, no. :shock:

First, they're assuming that you were given the perimeter value, rather than the values of the length and the width, which of course makes no sense, because you then wouldn't have any way to find the area of the bases. Second, they're jumping right to a shortcut (their favorite?), which leaves the reasoning hidden.

In effect, they've taken the prism and taken off the top and the bottom. These have the same area, so they've found the area of one of them, and then multiplied by 2. Then they took the rest, the "sides", if you will, cut along one of the edges, and unfolded. Laying this flat, you then have a length equal to what had been the perimeter (the distance around) and width equal to what had been the height. They found this area. Then they added the two values.

You should use whatever method you best understand, and always remember (especially in case you forget a formula) that the surface area of anything covered in polygons (like rectangles) can be found by finding the area of each face, and then adding.

Ok, I'd like to get this, the way they did it, but I am a little slow. If you have time to help. So, I see how they got the top and bottom area with the area times two. But with the sides. Ok, in the picture in the book the top and bottom length and width is 5 and 6. The side has a height of 7, so for those two sides it would be 7 times 6 to get the area. The 5 is only on the top and bottom. So they actually have gotten the area for the top and bottom, and added the perimeter of the base times the height. Seems like they are missing the side?

So, from what I am seeing their equation gives:

perimeter (of the base) times height (of the side) + 2 times area of base.

It seems like they are missing the sides (or I am not understanding)? I mean the area of the side would be 6 times 7.

What does multiplying a base times height give you? Maybe if I understand that it would help.

Sorry, I am sure you explained it well but I can't get my mind to understand what you were saying about taking the sides and cutting them. ha, I am new to this.

If any of that made sense.

Thanks.

I will post a picture of their diagram.