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### Course: Basic geometry and measurement > Unit 9

Lesson 2: Surface area with nets# Surface area using a net: rectangular prism

A polyhedron is a three-dimensional shape that has flat surfaces and straight edges. Learn whether or not a certain net could be folded up into a certain rectangular prism. Created by Sal Khan.

## Want to join the conversation?

- Is there a way to find the surface area without using a net? Isn't there a formula for doing this?(23 votes)
- Any prism is given by SA = PH +2B where P is the perimeter of the base (in a rectangular prism, you could choose any side as one of the bases), H is the height of the prism (the third dimension apart from the length and width of the base) and B is the area of the base. So if you can find the base (it could be a triangle, rectangle, parallelogram, etc.) and can calculate the area and perimeter, you can find the surface area. Cylinders work also, but C is used in the place of P and is 2 π r and B = π r^2, so S = 2π r H + 2π r^2 = 2 π r(H + r).(36 votes)

- how do you find the surface area of a hexagon(18 votes)
- Interesting question!

2-D figures, such as a hexagon, have area, not really surface area. 3-D figures have surface area.

A hexagon with side length s can be divided into 6 non-overlapping equilateral triangles with side length s.

An equilateral triangle with side length s can be divided into two right triangles, each with one leg s/2 and hypotenuse s, such that the legs with lengths s/2 together form the base of the equilateral triangle. It follows from the Pythagorean theorem that the leg common to both right triangles (the altitude of the equilateral triangle) is s*sqrt(3)/2. So the equilateral triangle's area is (1/2)s*s*sqrt(3)/2 = s^2*sqrt(3)/4.

So the area of a hexagon with side length s is 6s^2*sqrt(3)/4 = 3s^2*sqrt(3)/2.(20 votes)

- What is a formula? Can anyone give me an example? 🤔(10 votes)
- A formula is a way to express a relationship in math, for example the formula to convert from inches to feet is
*1/12 * number of inches*. In geometry, most formulas will probably look more like*2𝜋r*, the formula to get the diameter of a circle given the radius, or*base * height / 2*, the formula to get the area of a right triangle. I hope this helps! You can learn anything!(18 votes)

- what does surface area mean?(8 votes)
- Surface area is how many cubic units make up the surface.

Say a cube, for instance.

Take one face, and say it's 3 by 6.

The surface area of that one face would be 18 cubic units.(13 votes)

- So what is a POLYHEDRON?(7 votes)
- It's a 3-dimensional figure with multiple FLAT faces and NO curved surfaces. It's basically the 3D version of a polygon. An example of a polyhedron is a cube, while a cylinder is not a polyhedron (since it has curved faces).(8 votes)

- what if you do not have the little hash marks on the net?(6 votes)
- Check0:26for reference, its not gonna be the same area im pretty sure(2 votes)

- Why did people decide to use hash marks to show same lengths?(5 votes)
- They use it so you can see which ones are similar its also easy to know(2 votes)

- i have an question why do you have to shade it in and what is surface area mean(1 vote)
- To answer your first question, I think he does it so it is easier for us to understand what part he is talking about.

To answer your second question, the surface area is basically all the areas of each face combined.(5 votes)

- why does gooogle doodle not blocked at school(3 votes)
- What if it shows the line coming from the inside instead of the other way.(3 votes)

## Video transcript

- [Instructor] Teddy knows that
a figure has a surface area of 40 square centimeters. The net below has five-centimeter and two-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 five-centimeter edges. So this is one of the five-centimeter
edges right over here. And we know that it has several
other five-centimeter edges because any edge that
has this double hash mark right over here is also
going to be five centimeters. So this edge is also five centimeters. This is also five centimeters. This is also five centimeters. And then these two over here
are also five centimeters. So that's five centimeters
and that's five centimeters. And then we have several
two-centimeter edges. So this one was two centimeters. And any other edge that
has the same number of hash marks, in this case, one, is also going to be two centimeters. So all of these other edges,
pretty much all the rest of the edges are going
to be two 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,
and it looks pretty clear that this is going to
be a rectangular prism, but let's actually draw it. So if we were to... We're gonna fold this in,
we're gonna fold this that way. You could view this as
our base, right over here. We're gonna fold this in. We're gonna fold that up. And then this is going to be our top. This is the top right over here. This polyhedron is gonna
look something like this. So you're gonna have your base,
you're gonna have your base that has a length of five 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 can put the double hash marks
if I want, five centimeters, and that's of course the same
as that dimension up there. Now when we fold up, when
we fold up this side... Let me do this in orange, actually. When we fold up that side, that could be this side
right over here, this side right over here along
this two-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 five centimeters in this dimension. It is two centimeters,
two centimeters tall, and it is two centimeters,
two 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, and 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 gonna be five
centimeters times two centimeters. So it's gonna be 10 square centimeters. Same thing for this one. It's gonna be five by two, five by two. This one is five by two. So these are each 10 square
centimeters, and so is this one. This is five long, five centimeters long, two centimeters wide. So once again, that's
10 square centimeters. Now, these two sections right over here, they're two centimeters
by two centimeters, so they're each going to
be four square centimeters. So what's the total surface area? Well, 10 plus 10 plus 10 plus
10 is 40 plus four plus four 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.