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Lateral & total surface area of triangular prisms
Practice finding the lateral surface area and total surface area of a triangular prism using the formulas instead of nets. Created by Sal Khan.
Video transcript
- [Instructor] We're asked what
is the lateral surface area of the triangular prism, and what is the total surface
area of the triangular prism? Pause this video and try
to solve this on your own before we work through this together. All right, so first let's
just remind ourselves what lateral even means. In everyday life, lateral
means to the side, or if we're dealing with
the sides of something. Now, when we're thinking
about the lateral surface area of this rectangular prism, the
way it's oriented right now, it might not be exactly obvious
what they mean by the sides, the lateral surface area, and the general principle, when I'm thinking about lateral
surface area in particular, I like to think about, "Can I take this figure and
can I stand it up on a base?" And if you did, if you took
this one and you set it up so that this is the base right over here, well, then it might be a
little bit more obvious. Then, it will look something like this, where this is the base,
and then you have a height that looks something like
this, and this is the top, which has the same
surface area as the base. I know I could have, let me
draw it better than that. The best way is to just draw the top two and then connect these. So, it will look something like this. If we make this the base, where this seven is now the
height, this three is now the, I guess one we could think
about is the height of the base, that's three right over here. This length right over here is eight, and this length, right over here, is five. Now, when we're thinking
about surface area, and to be honest, my brain
likes to just go side by side and figure out all of the
surface areas and add them up, and that's probably how
I would've approached it if they weren't asking for
the lateral surface area, and if I didn't also
know that the standard we're trying to cover here, deals with formulas. You might sometimes see this
formula right over here, that lateral surface area is equal to perimeter of the base times height. Now, I'm not a big fan
of formulas like this, because you might just memorize
them and then forget them the next day, but it is useful
to know where this comes from and then we can actually apply it. So, one way to think about it, the lateral surface
area is the surface area of the three rectangles
that essentially connect the top here, this top triangle
and this bottom triangle, or if we're looking at this
orientation, it's this side, this side, and the
bottom, right over here. Now, when they talk about the perimeter, they're talking about, they're talking about this
length that I am showing in red, essentially the perimeter of the base, and that base is a triangle. Then, you multiply it by the height, and that makes sense, because
when you have the perimeter, you have this length, plus
this length, plus that length. If you wanted to find the
area of each of those panels, so to speak, you would multiply
the length of that side times its height, the length
of this side times the height, the length of that side times the height. Here, we're just adding
them all together first and then multiplying by the height. But let's just do that. What is the perimeter of
this base right over here? P is going to be equal to this eight. So, that is eight, and then this side over here is five, and then this side over here is also five. They tell us that. So, it's 8 + 5 + 5, which is equal to 18. That's the perimeter. And then what's the height here? Well, we see that the
height here is seven. So, that is the height. So, the lateral surface area is, let me just write it this way. It is equal to 18. That's the perimeter of the base, times the height, times seven. Let me do that in the same color. Times seven, which is going to be equal to 10 x 7 is 70, 8 x 7 is 56, 70 + 56 is 126. 126, and they don't
give us the units here, but we could say square
units or units squared, however we want to think about it. Now, another way you
could think about this is, if we were doing it separately, if we just did each panel by
themselves, you would first do, maybe you do this panel,
and you would say, "Well that area is going to be 5 x 7." And then, you would do
this panel over here, which is also going to be 5 x 7, and then you would do
this panel down here, which is essentially this
one, this big panel down here, which is 8 x 7, plus 8 x 7. This would also calculate
the lateral surface area. But notice, this is the same thing. If we were to just factor
out the seven here, this is the same thing as, this is the same thing as 7 x (5 + 5 + 8). Or the height, this seven times
the perimeter of the base. So, these are just equivalent things, but I'll get to this other part. What is the total surface
area of the rectangular prism? Well, for total surface area,
we just have to add the areas of the base and the top to
the lateral surface area. Sometimes, you'll see a formula like this, that the surface area is equal
to the lateral surface area, which is the perimeter of
the base times the height, plus two times the base, because that's the area
of the base, plus the top. I really, oh, sorry, this is a plus. I really don't like formulas like this, because you're going to
forget what they mean. It's better to just think
it through, common sense, but let's just do it where you
have the lateral surface area and now the surface area
of each of these triangles, well, base times height times one half. Area of a triangle is one
half times base times height. So, the base here is
eight, the height is three. So, area is equal to 1/2 x 8 x 3, 1/2 x 8 is 4, 4 x 3 is equal to 12 square units. So, that's one of them. We have two of them. We're going to have the
other one back there. So, if you say 126 + (2 x 12), so 24, that will get
us to 150 square units as the total surface area.