in this video you're going to learn how to find the surface area and the volume of three-dimensional figures we're going to talk about cones we're going to talk about pyramids we're going to talk about prisms and cylinders and spheres so let's get into this video and we're going to go through the formulas I'm going to show you an easy way to remember these formulas and how to group them together so that you don't forget them and you'll be able to solve these problems easily so let's get into the first group of problems the first group we're talking about prisms okay and basically what a prism is a prism is a figure a threedimensional figure where the two bases are parallel okay and they're congruent so that means they're the same size and shape and they're parallel they don't cross okay if they were to continue to go and they're separated by the height okay now I group cylinders okay with these prisms because a cylinder is really like a circular prism you can see the two bases are circles and they're separated by that height so when you think about cylinders and prisms think of them all as one group so we're going to do the surface area and the volume so first of all let's do the volume of this figure now when you do the volume the formula the general formula is very easy to remember it's the area of the base times the height so that's why I did a capital B so the area of the base means you find the area of the bottom that's what it sits on and you multiply by the height so essentially what this is like it's like taking the area of this Square in this case okay which is 16 in squared and it's like taking those squares like a stack of sticky notes and you stack them up okay 10 in high so you take the area of the base which is 16 time the height which is 10 which is going to give us 160 in cubed now the reason it's cubed is because you know we're filling these up with like little ice cubes you could think of okay uh something like like that right so little one by one by one cubes and that's how many are going to fill up the inside of this box right now let's talk about the surface area for a moment so the surface area is like if you were going to take a paintbrush and you were going to paint all the sides you know of this figure so when we think of the surface area this is the formula that you want to memorize it's 2B okay the capital b means the area of the base two of them because you have a top and a bottom plus the perimeter of the base times the height now let me see if I can show you a three-dimensional uh figure to help you to kind of understand this better so what I have here okay with us is basically a square prism okay so you can see that there's a square there at the top there's a square at the bottom okay okay so if we take the area of this square plus the area of this square that takes care of the two bases but now if you can kind of see here see the perimeter of this Square okay so you're follow me so if you take that perimeter if you unfold this figure that's the perimeter of that base okay so remember this was folded up like so okay but when we unfold it that's the perimeter of our base times the height and when you take the length times the width or the base times the height you're getting the area of this rectangle okay so you're with me so far so if I was to draw draw a net okay net is like if you were to unfold the shape what you would get is something that looks like this you would get four rectangles and you'd have a square here and a square here it's like taking a shoe box and unfolding it you might want to try that and so basically this Dimension right here is the perimeter of the base this Dimension here is the height and that's where we get the P * H that's what gives us the lateral area or the area of the sides then you just add the two bases the top and bottom okay so let's do this so basically the perimeter is going to be 4 plus 4 4 + 4 which is 16 times the height which is 10 okay so on my little diagram here this would be 10 and this would be 16 right so that's 160 okay plus the two bases now the base is a square so that's 4 * 4 is 16 * 2 since we have two of them so we have 32 + 160 which is 192 in squared okay now when you do area it's square units because you're covering the surface with these little one by one squares okay and that's how many would cover the entire outer surface okay so with me so far so now let's go over to this shape here try this one on your own if you can and uh basically what we're going to do is do the same thing this is a triangular prism the top and bottom are triangles they're separated by that height which you can see is 12 so let's start with the volume first so the volume is the area of the base times the height okay now because this is a triangle what you want to do is you want to think of the area formula for a triangle that's 1/2 little B * H so this is like our base here three this is like our height of our triangle here four so this is going to be 12 * 3 * 4 and then our height is going to be 12 okay so you don't want to get the heights confused this H is for the overall height the distance between the two triangles this H is really for the height of our triangle okay so that's why I recommend just memorizing the general formula then you can break it down into the sub formulas for each individual problem so let's go ahead and simplify this so this comes out to let's see 3 * 4 is 12 * 12 is 144 half of that is 72 and I didn't put the units here so I'm just going to say units cubed for the volume all right now let's go over to the surface area so surface area is really the total outer area right so what we're going to do is we're going to use the same formula that we were talking about here earlier okay which is the 2B plus pH so two bases plus the perimeter of the base times the height so so in this case the base is going to be a triangle so that's going to be 2 * 1/2 base * height plus the perimeter of the base times the overall height so let's see if we can do this so basically 2 * a half that's one so I'm just going to cancel those out the base of the triangle is three and the height of the triangle is four the perimeter of the base you have to add all these Up 3 + 4 + 5 which gives us 12 and the overall height is 12 so we get 144 + 12 which is 15 56 units squared because this is area so we're just covering this with these little squares now if you want to see a net okay like an unfolded uh picture of this diagram let me see if I can put that here for us it would look something like this okay you would have three four and five okay so when you unfold this and then you would have a triangle over here like this and a triangle over here like this so when you fold those that's going to be the top and the bottom so that's where we we're getting this uh 12 that's the perimeter okay times the height which is this height here that's 12 so that's how I'm getting 12 * 12 which gives you this area 144 then you just have to add the two bases the two triangles to get the total area Okay one more now we're going to talk about cylinders again a cylinder is like a circular prism the top and bottom are circles they're separated by that height so let's start with the volume so the volume is the area of the base times the height right so notice I'm using the same formula for all these prisms cylinders we're treating them as a group okay okay so the area of the base it's a circle so I'm going to write the sub formula for the base which is p piun r^ 2 time the height so in this case the radius of the circle is three so that's going to be 3^ squar the overall height is six so if we do this this is 3^ s is 9 * 6 is 54 Pi you can put Pi in your calculator 3.14 and multiply these together but I'm just going to leave it as 54 Pi M cubed and again that's how many little ice cubes that you are 1 m by 1 M by 1 meter that would fill this gigantic cylinder now if we want to do the surface area okay what we're going to do is we're going to again think of this as a a net so if we unfold this like if I was to take some scissors and cut that right there and unfold it you're going to have a circle here a circle here and you're going to have a rectangle here when you roll that rectangle let me see if I can show you with a piece of paper here it's going to look like this right so there's the circle at the top there's a circle at the bottom but when I enroll this there's our rectangle right there now when you look at this Dimension here okay this Dimension is really the circumference of the circle right so when I enroll that this is the circumference that's just another name for the perimeter okay of the base so we're going to use our formula 2B plus pH but remember the perimeter is really this Dimension okay that's like un rolling this uh Circle here like this so this is going to give us 2 pi r this is our height so that's where we get our perimeter 2 pi r time the overall height plus two bases remember the base is a circle so P pi r s so now all we have to do is substitute in the values so we've got 2 pi the radius is 3^ 2ar plus 2 pi the radius is 3 and the height is uh six and we just have to simplify so this would be 9 * 2 is 18 Pi this is 18 * 2 is 36 Pi if we add those together we get 54 piun M squared and it's just a a coincidence these both came out to 54 Pi this is me squared this is how many square meters would cover the outer surface of this cylinder so again think about volume and surface area you know just these three basic formulas B * H the area the base times the height and then the surface area formula 2B plus pH to find the outer area okay next we're going to talk about pyramids and cones let me erase this board and we'll uh start with those okay now we're going to talk about pyramids and cones and you want to group pyramids and cones together just to help you to narrow down the number of formulas you have to memorize and just make it easier overall so a pyramid and a cone you'll notice they both just have one base so they just have the one bottom unlike prisms that had a top and a bottom that were parallel and congruent these just have one base and you can see they go up to a point here okay this one uh vertex point at the top now when you look at a pyramid in a cone they've got two different uh things going on they've got this overall height okay which goes right down to the center of the base see this 12 right here and then they have something called the slant height or which they use the letter L for which I kind of call the Leaning height because it's on an angle like that so that's the slant height right there 13 and in this case the slant height is five so we're going to talk about those in these problems but first let's do the volume so the volume we're going to use this formula here 1/3 the area of the base times the height okay so what you do is you take the area of this bottom piece which is 6 * 6 that's 36 times 1/3 times the height now the height that you want to use is this overall height straight down to the center of the base in that case this one's four so this comes out to let's see uh 48 in cubed since it's volume all right so you're with me so far what this means this 1/3 area of the base times the height remember when we did the prisms okay like if I was to draw a prism like say for example like this okay just a rough sketch here what you can see is if these had the same base and they had the same height if you put this Pyramid inside of this prism you would actually be able to fit three of these inside of there okay so this one plus two more because you can see it's tapered it's you know angled up to a point like so so this actually only takes up a third of this volume so that's where this one3 is coming into play now let's go over to this one let's talk about the cone now so same formula volume is 1/3 area of the base times the height in this case the base is a Circle so we're going to use the formula P pi r s for a circle okay area of a circle times the height and again we want to use this overall height straight down to the center of the base that's 12 and let's see so for the circle the radius is 5 so that's going to be 5^ S is 25 okay now just a note here to you and that's that you know when you're uh multiplying these together multiplication is commutative you can change the order you're going to get the same result right so I'm going to take a thir of 12 which is 4 * 25 is 100 100 * pi so this is going to be 100 Pi units cubed and that's it you got the volume now let's talk about the surface area so the surface area the formula we're going to use for both the pyramid and The Cone is area of the base since you just have one bottom okay plus one2 the perimeter of the base times the slant height okay and it's the same formula over here volume is the area of the base plus 1/2 the perimeter times the slant height but what's different is it's a different shape base see this is a square so 6 * 6 is 36 right this one over here is a circle so that's going to beunk r^ 2 so < * 5^ 2 okay over here we're going to take the perimeter which is 6 plus 6 plus 6 plus 6 four sixes which is 24 and the slant height or the Leaning height which is five right but over here what we have is we have a circle so we have to take the perimeter of a circle which is the circumference which remember the formula for circumference is 2 pi r so 2 pi * 5 times the slant height so that leaning height which is 13 so when I think of the L I think of the the Leaning height right now all we have to do is go back and simplify so we've got half of uh 24 which is 12 12 * 5 is 60 and we just add the 36 and the 60 together to get 96 in squared so remember Square for area that's two-dimensional cubed for volume that's threedimensional for this one we get 5^2 is 25 Pi uh let's see 2 * 5 is 10 * a half is 5 that's 5 Pi * 13 is 65 PI right and if we add those together what do we get we get 80 uh 90 Pi uh unit squared now notice I just left the pi in there that's an exact answer but if you want to get an approximation you can put 3.14 in for pi multiply by 90 and you got it so again for pyramids and cones you want to think of the two basic formulas which is the volume formula see it's the same 1/3 area the base times the height and for the surface area I wrote volume here this is surface area Okay the outer surface okay one base plus 1 12 PL so it's just the area of the one bottom plus 1/2 the perimeter uh times the slant height so let me erase this board and then we're going to talk about spheres but the key is to group these together to make it easier you know to memorize these formulas okay lastly we're going to talk about spheres but before I get into spheres I just wanted to mention that if you're preparing for the act or the SAT Math section and you want to boost your score I've got two courses available you can uh check them out I've got links on my about page I'll put links in the description below but basically I've got uh one for the SAT which goes over 39 concept areas that are important to really you know boost your score on the test and then with the ACT Math section I've got 65 Concepts that really go into depth with teaching and examples and uh practice problems and so forth to really help you to booster score so if you like my teaching style check out those courses a lot of students have benefited from them and I'm sure you will too but let's talk about spheres now so a sphere is basically like a circle in three dimensions you know it's the set of all points that are equidistant from a given point which is called the center so it's basically like a ball like a basketball right and so if we're trying to find the volume like the Inner Space the three-dimensional space we're going to be using this formula here it's 4/3 pi * the radius cubed so there's really just that one dimension the radius now sometimes I'll try to confuse you a little bit maybe by giving you the diameter but you can always cut it in half to get the radius so for this one we're just going to say 4/3 < * 3 cubed 3 cubed is 3 * 3 * 3 which is 27 so we have 4/3 * < * 27 which is like 27 over 1 and you can do a little bit of cross reducing numerator and denominator here so that ends up coming out to multiply the numerators and denominators together that gives you 36 Pi units cubed that's the volume right now for the surface area the formul is a little bit different it's 4 pi r R 2 okay and what we want to do here is just put in our radius which is three so that's 3^ s is 9 * 4 is 36 Pi uh unit squared since it's area that's the outer surface now it's just a coincidence we came out with 36 for both of these uh but the formula is different you won't always get that that situation so again just trying to show you an easy way to group these together memorize the general formula and then you can break it down for each specific shape whether it's a triangular pyramid or it's a square pyramid or it's a cone or whatever type of shape it is so I hope you enjoyed this video subscribe to the channel check out over the over 400 other videos I have on my Mario's math tutor YouTube channel to help you boost your score in your math class improve your understanding and make math uh learning math a lot less stressful and uh I look forward to seeing you in the future videos I'll talk to you soon