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akira receives a prize at a science fair for having the most informative project her trophy is in the shape of a square pyramid and it's covered in shiny gold foil so this is what her trophy looks like how much gold foil did it take to cover the trophy including the bottom and so they give us some dimensions and we want how much gold foil and it's an inches squared so it's really going to be an area so pause this video and see if you can figure that out how much gold foil did it take to cover the trophy all right now let's work through this together and so essentially what they're asking is what is the surface area of this square pyramid and we're going to include the base because that surface area is how much it's the area of the gold foil that is needed now some times some of you might be able to think about this just by looking at this figure but just to make sure we don't miss any area I'm going to open up this square pyramid and think about it in two dimensions so we're going to do is imagine if I were to unleash or if I were to cut the top and let me just in red if I were to cut this edge if I were to cut this edge if I were to cut that edge and that edge so the edges that connect the pit be triangular sides and if I were to just open it all up what would this look like so if I were to open it all up well at the bottom you would have your square base let me color that in so you have your square base so let me draw that so you have your square base it's going to be a rough drawing and what are the dimensions there it's three by three we know this is a square pyramid so the base all the sides are the same length they give us one side but then if this is three inches and this is going to be three inches as well let me color it that same color just so we recognize that we're talking about this same base and if we open up the triangular faces what's it going to look like well this is going to look like this this is a rough sand drawing but hopefully it makes sense this is going to look like this and each of these triangular faces they all have the exact same area and the reason why I know that they all have the same base 3 and they all have the same height 6 inches but I'll draw that in a second so they all look something like this just hand drawing it and all of their heights all of their heights are 6 inches so this right over here is 6 inches this over here is 6 inches this over here is 6 inches and this over here is 6 inches so to figure out how much gold foil we need we're trying to figure out the surface area which is we're just going to be the combined area of these figures well the area of this central square is pretty easy to figure out it's 3 inches by 3 inches so it would be 9 inches 9 inches squared now what are the area of the triangles well we could figure out the area of one of the triangles so they're multiplied by 4 since there are 4 triangles so the area of this triangle right over here it's going to be 1/2 times our base which is 3 times 3 times our height which is 6 let's see 1/2 times 3 times 6 it's 1/2 times 18 which is equal to 9 9 square inches or 9 inches squared so it's going to be our total area well you have the area of your square base plus you have the four sides which each have an area of 9 so I could write it out I could write 4 times 9 or I could write 9 into that black color or I could write 9 plus 9 plus 9 plus 9 and just to remind ourselves this is that right over there is the area of one triangular face triangular triangular face so this is all of the triangular faces triangular faces and of course we have to add that to the area of our square base so this is 9 plus 9 times 4 you could view this as 9 times 5 which is going to be 45 square inches nine plus nine plus nine plus nine plus nine