Pre-Algebra: Surface Area of a Prism

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SURFACE AREA OF A PRISM

Volume of a three dimensional object tells how much material is needed to make the whole solid. What if we just look along the surface? If you think about it, we take a three-dimensional solid, and we see that the surface is actually two-dimensional. In fact, for this rectangular prism, we see the surface, as we know, has two rectangular bases, and four rectangular faces. If we want to imagine building just the outside region of this, the amount of surface required can be found by computing the surface area. The surface area is the total area of all the surface on the outside of a particular object.

For example, if you wanted to find the surface area, what you would do is find the area of each face and add them all up. For example, let’s consider this rectangular prism. It is a little bit rickety, because I haven’t glued it together. You can see that this is a rectangular prism. It is like a box. How would you find the surface area of that? Well, maybe the easiest way to do that is to unfold the box, which is easy to do. We produce this net. So really this represents 1, 2, 3, 4, 5, 6, the 6 faces of the box. You can see each one is a rectangle. To find the surface area, all I have to do is find the area of each rectangle, add them all up, and that is the surface area of the rectangular prism.

Let’s look at an example together. For this one, I am told that this is 3 centimeters and this right here is 10. Let’s make sure that we are all clear that this is up to here. This from here to here is 10. This is 5. This is 5. This is 10, and so forth. If we want to find the surface area, we just find the area of each rectangle and add them all up. So, 10 times 3 is 30 for area. Here this rectangle is going to be 3 times 5 which is 15. This is 5 time 10 which is 50. This is 3 times 10 which is 30. This is 5 times 3 which is 15. This is 5 times 10 which is 50. If we add all of these up, what do we see? We see that 50 and 50 is 100. 30 and 30 and 30 is 90. So we see 190. Now, are the units centimeters cubed? No because this is not volume, this is area. Remember that in each case it was base times height, centimeters squared. Surface area has units squared. So, for a rectangular prism, we can find it this way.

Notice that if we look at it systematically, we had the bases of the box. Right? Those are going to be the two exact same rectangles. We take the area and multiply it by 2. Two opposite sides have the same area. We multiply that by 2, and the other sides by 2. Really, we can write the surface area for this shape pretty easily in general, but by saying you just find the area of one of these rectangles and multiply by 2, the area of one of these rectangles and multiply it by 2, and then the area of one these rectangles and multiply by 2. It is just 2 times length times width plus 2 times length times height plus 2 times width times height. That is all there is to it. Pretty cool.