If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

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?

  • ohnoes default style avatar for user Thundermijo
    Is there a way to find the surface area without using a net? Isn't there a formula for doing this?
    (15 votes)
    Default Khan Academy avatar avatar for user
    • mr pink green style avatar for user David Severin
      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)
  • blobby green style avatar for user Hannia Tapia
    how do you find the surface area of a hexagon
    (12 votes)
    Default Khan Academy avatar avatar for user
    • primosaur seed style avatar for user Ian Pulizzotto
      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)
  • starky sapling style avatar for user 🐺Kendra Hammonds🐺
    What is a formula? Can anyone give me an example? 🤔
    (6 votes)
    Default Khan Academy avatar avatar for user
    • leafers ultimate style avatar for user Neil Gabrielson
      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)
  • leaf grey style avatar for user Alanis Gomez
    what does surface area mean?
    (5 votes)
    Default Khan Academy avatar avatar for user
  • orange juice squid orange style avatar for user J_McK
    So what is a POLYHEDRON?
    (5 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user makenzie.stewart281
    what if you do not have the little hash marks on the net?
    (5 votes)
    Default Khan Academy avatar avatar for user
  • marcimus red style avatar for user Nick
    i have an question why do you have to shade it in and what is surface area mean
    (2 votes)
    Default Khan Academy avatar avatar for user
  • ohnoes default style avatar for user Silas Lundquist
    Why did people decide to use hash marks to show same lengths?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • leafers ultimate style avatar for user Dominic Palomo
    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)
    Default Khan Academy avatar avatar for user
  • starky seedling style avatar for user chloe :3
    How do you find a surface area of a octagon
    (2 votes)
    Default Khan Academy avatar avatar for user

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.