Finding surface area: nets of polyhedra | Perimeter, area, and volume | Geometry | Khan Academy

Finding surface area: nets of polyhedra | Perimeter, area, and volume | Geometry | Khan Academy


Teddy knows that a
figure has a surface area of 40 square centimeters. The net below has 5 centimeter
and 2 centimeter edges. Could the net below
represent the figure? So let’s just make sure we
understand what this here represents. So it tells us that it
has 5 centimeter edges. So this is one of
the 5 centimeter edges right over here. And we know that it has several
other 5 centimeter edges because any edge that
has this double hash mark right over here is also
going to be 5 centimeters. So this edge is
also 5 centimeters, this is also 5 centimeters,
this is also 5 centimeters, and then these two over
here are also 5 centimeters. So that’s 5 centimeters,
and that’s 5 centimeters. And then we have several
2 centimeter edges. So this one has 2 centimeters. And any other edge that has
the same number of hash marks, in this case one, is also
going to be 2 centimeters. So all of these other
edges, pretty much all the rest of the edges,
are going to be 2 centimeters. Now, they don’t ask us to
do this in the problem, but it’s always fun to
start with a net like this and try to visualize the
polyhedron that it actually represents. It looks pretty
clear this is going to be a rectangular prism. But let’s actually draw it. So if we were to– we’re
going to fold this in. We’re going to
fold this that way. You could view this as
our base right over here. We’re going to fold this in. We’re going to fold that up. And then this is
going to be our top. This is the top right over here. This polyhedron is going to
look something like this. So you’re going to
have your base that has a length of 5 centimeters. So this is our base. Let me do that in a new color. So this is our base
right over here. I’ll do it in the same color. So that’s our base, this
dimension right over here. I could put the double
hash marks if I want. 5 centimeters, and
that’s of course the same as that
dimension up there. Now, when we fold up this
side– we’ll do this in orange, actually– when we
fold up that side, that could be this side
right over here, along this 2 centimeter edge. So that’s that side
right over here. When you fold this side in right
over here, that could be that. That’s that side
right over there. And then when of course we
fold this side in– that’s the same color. Let me do a different color. When we fold this
side in, that’s the side that’s kind of
facing us a little bit. So that’s that right over there. That’s that right over there. Color that in a
little bit better. And then we can
fold this side in, and that would be that side. And then, of course,
we have the top that’s connected
right over here. So the top would go–
this would be the top, and then the top
would, of course, go on top of our
rectangular prism. So that’s the figure
that we’re talking about. It’s 5 centimeters
in this dimension. It is 2 centimeters tall,
and it is 2 centimeters wide. But let’s go back to
the original question. Is this thing’s surface
area 40 square centimeters? Well, the good
thing about this net here is it’s laid out all
of the surfaces for us, so we just have to figure
out the surface area of each of these sections and then
add them together, the surface area of each of these surfaces. So what is the surface
area of this one here? Well, it’s going to be
5 centimeters times 2 centimeters. So it’s going to be
10 square centimeters. Same thing for this one. It’s going to be 5 by 2, 5 by 2. This one is 5 by 2. So these are each 10
square centimeters, and so is this one. This is 5 long, 5 centimeters
long, 2 centimeters wide. So once again, that’s
10 square centimeters. Now, these two sections
right over here, they’re 2 centimeters by 2 centimeters. So they’re each going to
be 4 square centimeters. So what’s the
total surface area? Well, 10 plus 10 plus
10 plus 10 is 40, plus 4 plus 4 gets us to
48 square centimeters, or centimeters squared. So could the net below
represent the figure that has a surface area
of 40 square centimeters? No. This represents a figure
that has a surface area of 48 square centimeters.

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