[Music] mm ooh math today we are going to work some problems on volume and surface area this is part of the seventh grade year in review so this one is finding the volume of a right triangle er prism so we have a right triangle er prism that measures 4 by 10 by 15 so we have one side that's 4 meters one side that's 10 meters and we have a right angle right between those and the hypotenuse is 15 we also have the height of this prism which is 14 meters so now how do we find the volume well the first thing you're going to do is you're going to find the surface area or the area of the bases well the bases are the two sides that are parallel to each other and in this one those are the triangles so we have to find the area of a triangle so I'm going to come down here and redraw it this is 4 and this is 10 so we're going to use these two legs to find the area and the area of a triangle is 1/2 the base times the height so 1/2 in this case the base is 4 and the height is 10 so 1/2 of 4 times 10 that's 20 square meters and that's the surface area of the base then I take that surface area of the base that base area and I multiply it times the height so this is now 20 and our height is 14 so 20 times 14 is 280 and that's now volume so I've taken a meter squared times a meter and that makes it meters cubed and that is how we find the volume of this triangular right triangular prism now we're going to take that same right triangular prism and we're gonna do something a little bit different with it we're gonna find its surface area so we need to analyze this shape again but thinking of it in terms of just the surface area so the surface area we're gonna find the area of both ends of the triangle well we just found those areas we drew that triangle here and we said it was 4 by 10 well what's the area of a triangle it's 1/2 the base times the height when we said 1/2 of 4 times 10 that is gonna give us 20 now that's the surface area of one of these triangles but guess what I have two of those one on either end of my prism so I'm gonna take that 20 and I'm going to double it so that surface area of my two triangles is 40 square meters now I need to look at the what we call the lateral area for the sides of the prism okay to find that area we're gonna add the perimeter of the base so it's the perimeter of the base times the height of the prism so let's find the perimeter the perimeter is 4 plus 10 plus 15 so 4 plus 10 plus 15 and that's gonna give us 29 and that's the perimeter of the base and I'm gonna take that I'm gonna multiply it times the height of 14 now that one's not as easy to do in my head so I'm gonna grab my calculator and I'm gonna take 14 times 29 that gives me 406 so that's four hundred and six square meters now I have the two end caps these two ends there are 40 so let's add those together so 4 Oh six plus 40 gives me 446 square meters and that's how you find the surface area of a triangular prism so let's recap that you're going to find the area of the two triangles so find the area of one and double it and then take your perimeter times the height and that gives you the lateral area and you add those together to find the surface area so those are the first two now we're gonna switch shapes now we're going to try to find the surface area of a square pyramid now the square pyramid I'm gonna draw what's called a net okay now what a net is it's taking that three-dimensional object and laying it flat so if I lay this pyramid flat on the ground I have a square in the middle and these four sides lay down as three as four triangles so what I can do is I can find the area of each shape I can find the area of my square and the area of one of my triangles and then just multiply that by four so let's break this down okay the area of the square is pretty easy it's just an eight by eight square base so the square part it's just going to be 64 square inches now let's look at our triangle our triangle the face of it has a height of 10 the base of it was already given as 8 so I'm going to take to find the area of one of my triangles 1/2 the base which is 8 times the height which is 10 and that's 80 half of 80 is 40 and 40 represents the area of one triangle but I have four of those so I'm gonna take 4 times 40 to get the area of my triangles and that's 160 now I'm going to add the square to the four triangles let's see four that's a 12 carry the 1 so that's 224 square inches for my surface area and again I think drawing the net which is that shape just taken and flattened out is the easiest way to look at that ones okay now we're gonna look at that same shape but it says if the square pyramid from number three which is the last problem was painted on the outside not including the floor what would be the surface area of the painted area okay so what would be the area if we took the floor out well our answer in the last problem was 224 square inches well if we want don't want to paint the square bottom we're going to take that area and subtract it so what are we going to subtract we're going to subtract the area of the square for the area of the square was 8x8 or just 64 so what's 224 minus 64 that ends up being 160 square inches which is the area of the four triangles that we added up so there's number 4 now let's look at the surface area of a triangular pyramid okay so a pyramid always has a point at the top so this is different than a prism because the prison has two parallel bases this one has what we call an apex which I'm trying to do so you can see has a point at the top so this one has a point at the top so that's why this one's a pyramid and not a prison so what we're going to do is we're going to find the surface area of this triangular prism and because each side is different I'm gonna have to work a little bit harder to find this surface area okay so let's find the area of each side separately so this first side I'm going to draw this this side I'm going to highlight it right here this is 8 is the base and the height is right here and it's given as 22 so I'm going to draw that over here so 8 and 22 so to find that area I'm just gonna take 1/2 the base times the height 1/2 of 8 times the height of 22 so let's take that as 0.5 which is 1/2 times 8 times 22 and that gives me 88 square inches for that first side okay now let's look at this side over here okay this side has a side of 23 and a base of 10 when we don't know the surface area of we don't know this height so we're gonna have to work a little bit to get it okay so this is 10 and this is 23 and we don't know this height so I'm going to draw an altitude down and we're gonna use 23 and that means this side has to be 23 this is an isosceles triangle so that's gonna split that into 5 and 5 and then I've got to use the Pythagorean theorem to find this height so let's take a squared which is 5 squared plus the B squared which is our height that we don't know is equal to C squared which is 23 squared so let's see 23 we're gonna square that we're gonna subtract 5 squared and that's going to be 504 and I'm now going to take the square root of 504 and that gives me 21 or 22 point 4 so that is my height 12.4 now I can find the area 1/2 the base is 10 and the height is 22.4 so let's multiply that together 0.5 times ten times twenty two point four that gives me 112 square inches so now we found two of the sides now we're gonna have to work even harder so over here I've now got this third lateral face I don't know the base and I don't know the height but I can do a little bit of math to figure it out if you look the base of this triangle right here is a right triangle there we're given a right angle so this is eight and this is ten I can use the Pythagorean theorem to find this base so let's work on that first base so 8 squared plus 10 squared those are my two legs that's 164 I'm now going to take the square root of that and that gives me 12 point eight okay I have a side of 23 and another side of 23 I've got this side over here was 23 and the sides 23 so I can now drop an altitude down and say this is now 6.4 it's half of it 6.4 and I can use the Pythagorean theorem to find the height so 23 is my hypotenuse minus 6.4 squared that's 23 squared by the way and that's 4 for 88 let's take the square root of 4 88 and that will give me my height of 20 2.09 now I have enough information to find this area one-half the base which is twelve point eight times the height of 20 2.09 so 0.5 times twelve point eight times twenty two point zero nine and that gives me a 144 or 140 1.4 square inches okay so that gives me all three lateral faces all I have left is to find the area of the base well guess what that one's easy you know why it's a right triangle and this is 8 and this is 10 so 1/2 the base of 10 times the height the bait that's gonna give me 40 square inches so now what do I do with each of the triangles to find surface area I add them up so I'm gonna circle them the first one was 88 the second one was 112 the third one was 144 and that little base was 40 so let's add those 4 up 88 plus 112 plus one 41.4 plus 40 and that gives me a grand total of three hundred and eighty one point four square inches for the surface area of that triangle that was definitely a tough one okay now let's move on find the surface area of a cube with a side length of seven okay so we're looking at surface area so let's first draw a picture a picture's worth a thousand words so you want to draw your self a cute sided cube and each side is seven so the length width and the height they're all seven now to find the surface area that's pretty easy if you think about a dice how many sides does a cube have it has six sides what each side has the same area so what's the area of one of these faces well it's a square that's seven by seven so one square is 49 square units square centimeters in this case so I'm gonna take six faces times 49 and that's gonna give me my surface area so 6 times 49 oops it's clear that that's going to give me 294 so two hundred and ninety-four square centimeters is the surface area and how did I do that I found the surface area of one side and I just multiplied it by the six sides and there you go there's your review of surface area and volume [Music]