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## 6th grade

### Unit 10: Lesson 4

Nets of 3D figures# Surface area using a net: rectangular prism

CCSS.Math:

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?(15 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).(22 votes)

- how do you find the surface area of a hexagon(12 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.(11 votes)

- What is a formula? Can anyone give me an example? 🤔(6 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!(11 votes)

- what does surface area mean?(5 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.(9 votes)

- So what is a POLYHEDRON?(5 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).(6 votes)

- what if you do not have the little hash marks on the net?(5 votes)
- Check0:26for reference, its not gonna be the same area im pretty sure(1 vote)

- i have an question why do you have to shade it in and what is surface area mean(2 votes)
- 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 did people decide to use hash marks to show same lengths?(3 votes)
- With triangles do we do the same formula? I am asking this because when I tried this formula with triangles it did not work I ended up with a bad grade on a test, so I wanna know why this formula didn't work so I can get better with the questions with triangles, any response answering this question would help.

Written by: Dominic Palomo(⌐■_■)(3 votes) - How do you find a surface area of a octagon(2 votes)
- Also, you can never get the actual surface area of a hexagon (2 is irrational.(1 vote)

## Video transcript

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.