Surface Area of a Pyramid & Volume of Square Pyramids & Triangular Pyramids

in this video we're going to talk about how to find the surface area and the volume of a square based pyramid and a triangular pyramid so let's draw a picture of the square based pyramid just a rough sketch and let's calculate the volume of this pyramid first so the line in red is basically the height of the pyramid so let's say the pyramid has a base length of 6 and the height of the pyramid let's say it's 10 units long how can we find the volume of this pyramid to find the volume you need to use this equation it's one-third times the area of the base multiplied by the height of the pyramid the height in this example is h and we need to find the area of the base let's call this b but low case b you can call it s or x if you want to depending on if that's the way your teacher taught it or if it's in your textbook like that but i'm going to use lowercase b for the side length of the base and capital b for the area of the base so to find the area of the square it's going to be the length times the width or simply b squared so the area of the base is b squared so therefore the volume of this entire square base pyramid is 1 3 times the side length squared multiplied by the height of the pyramid so it's going to be 1 3 6 squared times ten six times six is thirty six and one third of thirty six or thirty six divided by three that's twelve and twelve times ten is a hundred twenty so the volume is 120 cubic units so let's say if this was 6 inches the volume would be 120 cubic inches if everything was in inches let's try another practice problem so go ahead and find the volume of this pyramid so let's say the side lamps are eight inches and the height let's say it's 12 inches so feel free to pause the video and find the volume of this pyramid so the volume is going to be one-third times the area of the base multiplied by the height of the pyramid and the area the base as we defined in the last example is going to be the side length of the square squared so basically just b squared so now example b lowercase b is eight and the height is twelve so it's one third multiplied by eight squared times twelve one-third of 12 or 12 divided by three that's four and eight squared is sixty-four four times sixty-four is two hundred and fifty-six so the volume is 256 cubic inches and so that's the answer for this example now let's talk about finding the slant height of a square based pyramid so let me just draw this first i want to draw this carefully now the red line is the height of the pyramid as we talked about before so this makes a right angle with the plane of the pyramid the line in green is the slant height so make sure you're aware of the difference between these two so this is l which represents the length of the green dashed line and the red dashed line that represents the height of the pyramid we're going to say b represents the side length of the square now we've talked about the volume of a pyramid it's 1 3 b squared times h or one third the air of the base times h but now if you want to find a surface area you need to use the slant height instead of the actual height the surface area is the area of the base plus the lateral area now the area of the base is simply the area shaded in blue and basically that's just b squared now what about the lateral area notice that this pyramid has four triangles here's one of the triangles shaded in blue so that's just the triangle on the right side you have a triangle on the left one in the back and one in the front so that triangle i'm just going to redraw just a side view of it so this is the base length of the square which is also the base of the triangle and the line in green represents the slant height of the triangle which is l now whenever you have a triangle if you know the base and if you know the height of the triangle to find the area is simply one half base times height but in this case the height is this line height so the area becomes one half base times slant height which is l now the triangular pyramid has four triangles you have one on the left highlighted in green and then you have one in the back which is the blue line's kind of overlapping it but you can see in the back there and then the one in front represented by the portion shaded in yellow so therefore it's going to be four times the area of each of those triangles which is one half b times l so four times a half is two so therefore the surface area you could find it using this equation it's b squared plus 4 i mean 2 bl so that's the formula i'm going to use to find the surface area of a square base pyramid it's the area of the base plus the lateral area go ahead and try this example so let's say if you're given the slant height and let's say it's 15 centimeters and you're given the length of the base let's say it's eight centimeters it's eight by eight go ahead and calculate the surface area of this object or figure and also find the lateral area as well now the first thing i would do is find the lateral area the lateral area is simply 2bl as you mentioned before it's one half base times height or in this case i that's the area of each triangle and there's four triangles so times four and that gives us 2bl so that will give you the lateral area the area of the four triangles so b in our example is 8 and l is fifteen so it's two times eight times fifteen now two times fifteen is thirty and thirty times eight if three times eight is 24 30 times 8 is 240. so it's 240 square centimeters so that's the lateral area now what we need to do next is we need to find the surface area it's the area of the base plus the lateral area so it's going to be b squared plus 2bl b squared that's going to be 8 squared and 2bl is 240. 8 squared is 64. 64 plus 240 that's going to be 304 so the surface area is 304 square centimeters and so now you know how to find the surface area and the lateral area of a square based pyramid so these are the two equations that you need now let's work on another problem let's say the height is 12 inches so that's going to be the height of the pyramid and let's say the base is 10 inches long and 10 inches wide so using this information find both the volume and the surface area of this pyramid so be careful this problem so go ahead and take a minute and work on this problem feel free to pause the video too so let's start with the volume it's one third base times height which is one third the side length squared times the height of the pyramid b is 10 and the height is 12. so 1 3 of 12 is 4 and 10 squared is 100 so in this problem the volume is 400 cubic inches so that wasn't bad now what about the surface area and also the lateral area how can we find those things well first we need to calculate the slant height so we got to find the length of this green line so how can we do that notice that we can make a right triangle so i'm just going to redraw the right triangle we have the height of the triangle we know it's 12. we need to find the length of the green line which is l this height but what is the base of the triangle well if b is 10 and we know that the red line the red dotted line has to be at the center of the pyramid therefore this section must be half of 10. so if this whole thing is 10 then this portion must be 5. it has to be half of 10 because this is the midpoint of the pyramid so now we can use the pythagorean theorem to find the slant height so for those of you who want an equation for this process since b is 10 5 is b divided by 2 it's half of the base length so to find l for the square base pyramid l squared is equal to b over 2 squared plus h squared so you can use that formula if you ever need to find the slant height if you know the length of the base and the height of the pyramid so b over 2 is 5 and h is twelve five squared is twenty-five twelve squared is one forty-four when added together that will give you one sixty-nine so now we gotta take the square root of both sides the square root of 169 is 13. so that's the length of the slant height so now that we have it we could find the lateral area which is simply 2 b l so it's two times ten times thirteen now ten times thirteen is one thirty and two times one thirty is two sixty so that's the lateral area it's 260 square inches now the last thing that we need to do for this problem is calculate the surface area the surface area is the area of the base plus the lateral area which is b squared plus 2bl so b squared that's going to be 10 squared plus 2 times 10 times 13. so 10 squared is 100 and we know that 2 times 10 times 13 that's the lateral area which is 260 giving us a total surface area of 360 square inches so that's the surface area of this figure so just be careful when you need to calculate the slant height and as long as you know how to find it finding the volume surface area and lateral area should be a piece of cake now let's talk about the triangular pyramid so i'm going to have to draw this one carefully i'm going to draw a nice big picture so you can clearly see everything and this is a right angle and in red just like i did before this is going to represent the height of the actual pyramid the line in red is the height of the pyramid just like before now this section here is the base of the triangle and this section here we're going to call it the height of the triangle so in the last pyramid that we dealt with we had a square base pyramid but here this is a triangle base pyramid so we need the base and height to find the area of that that base the triangular base now to find the volume is one-third times the area of the base multiplied by the height so in order to find the area of the base which is this triangle at the bottom we need to use this formula it's one half base times height so make sure you distinguish the height of the pyramid which i'm using capital h versus the height of the triangular base which i'm going to use as a lowercase h to distinguish them so this will give you the volume of this particular pyramid so let's work on an example so feel free to pause the video and try this problem so let's say this is 5 this is 13 this side is 12 and this portion the height of the pyramid is 15. so with this information go ahead and calculate the volume of the pyramid so this is 15 that's the height of the pyramid 5 is the base of the pyramid 12 is the height of the base of the triangle so now let's use the formula volume is 1 3 the area of the base multiplied by the height of the pyramid so that's one-third times the area of the triangle which is one-half base times height multiplied by the height of the pyramid so the base of the triangle is five units long the height of the triangle is 12 units long and the height of the pyramid is 15 units so let's see how we can do the math here one half of 12 is six so now we don't have to worry about those two numbers anymore 1 3 of 15 or 15 divided by 3 that's 5 and we still have another 5 to deal with now 5 times 6 is 30 and 30 times 5 is 150 so the volume is 150 cubic units so now you know how to find the volume of a triangular pyramid now let's focus on finding this surface area of a triangular pyramid so let's draw a picture first and let's say the line in green is the slant height so we have the slant height l we have the base of the triangle and the height of the triangle now most examples that you'll see when finding the surface area is that the triangular base is usually an equilateral triangle in order to simplify the calculations otherwise this problem might be more difficult than for a typical pre-algebra course so if it's uh equilateral triangle then instead of having a height this is also going to be b it's going to be a base length and this 2 would be b as well and for most examples that you will see for this type of problem it's usually an equilateral triangle now to find the surface area it's going to be the area of the base plus the lateral area now the area of the base the area of an equilateral triangle it's the square root of three over four times b squared that's for an equilateral triangle and the lateral area there are three triangles that we need to cover so this is the triangle on the right side this is the triangle in the back and we have a triangle in the front now sometimes they may give you the area of the base if that's the case you could just replace b with that number but if you need to find the area of the equilateral triangle you can use this formula now there's three lateral faces and the area of each of those faces is one half base times height but we need to use the slant height instead so it's one half b times l now sometimes you might be given the height of this triangle if you're given the height of the triangular base then you don't need to use this formula instead the area of the base will now be one half base times height so depending on what you're given sometimes you could use this formula one half base times height plus three over two b times l if you combine three times a half so sometimes you might be using this formula whereas other times you may need to use this formula depending on what's given to you let's try an example so let's say the left for the red line we're going to say it's a 10.4 inches and the base of that triangle is going to be 12 by 12 by 12. so it's an equilateral triangle and let's say that the slant height let's say it's 15 inches long using this information and go ahead and calculate the surface area of this triangular pyramid so what we need to do is find the area of the base plus the latter area now since we have the height of the triangular base we can use this equation to calculate the area of the base we can also use this equation too root 3 b squared over 4 since we have an equilateral triangle in fact let's do it both ways so you can see that you'll get the same answer now the lateral area we said it's three times one half base times slant height now the base is 12. the height of the triangular base that's 10.4 l is 15. so now half of 12 is 6 and then 6 times 10.4 that's 62.4 so that's the area of the base now to find the lateral area it's basically 3 over 2 b times l so 3 times 12 that's 36 divided by 2 that's 18 times 15 that's 270. so if we add 62.4 to 270 that's going to give us the surface area which is 332.4 square inches so that's how you could find the surface area in this problem now if we were to use this formula let's see what we're going to get so b is 12. b squared 12 squared is 144 divided by 4 that's 36 so this is going to be 36 root 3. 36 root 3 is 62.35 which if you round it that's about 62.4 so you should get the same value so this height is not an exact value it's a rounded value but the answer for this problem is 332.4 that's how you could find the surface area used in that equation let's try one more problem so let's say if we're given a slant height of 20. and we're given the base left let's say it's eight and units are all centimeters for every dimension go ahead and calculate the surface area and the lateral area of this object so let's start with the lateral area the lateral area we know it's going to be three times one half base times the slant height so the length of the base is eight and the slant height is twenty now half of eight is four and four times twenty is eighty and three times eighty is two forty so the lateral area is two 240 square centimeters now let's find the surface area the surface area is going to be the area of the base plus the lateral area which we already have now this time we don't have the height of the triangle or the triangular base so in order to find the area of this equilateral triangle we need to use this formula square root 3 over 4 b squared so b is 8 and the lateral area is 240. 8 squared is 64 and 64 divided by 4 is 16 so right now we have this the surface area is 16 root 3 plus 240 so that's the exact answer but if you want to get a decimal value multiply 16 by the square root of 3 and that should give you 27.7 if you round it to nearest tenth and then let's add 240 to it so the surface area is 267.7 square centimeters so that's the answer you

Channel: The Organic Chemistry Tutor

Views: 494,886

Keywords: surface area of a pyramid, volume of a pyramid, area, lateral area of a triangular pyramid, lateral area of a square pyramid, surface area of a square pyramid, surface area of a triangular pyramid, volume of a square pyramid, volume of a triangular pyramid, square pyramid, triangular pyramid, surface area, volume, lateral area, formula, explained, geometry, basic geometry, examples, problems, practice problems

Id: x8wEnG4GURQ

Length: 29min 41sec (1781 seconds)

Published: Mon Jul 31 2017