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 square based pyramid and triangular pyramid. So, let's draw picture of the square based pyramid. Just 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 base length of six 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 1/3 times the area of the base multiplied by the height of the pyramid. The height in this example is And we need to find the area of the base. Let's call this but lowerase You can call it or if you want to depend 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 for the side length of the base and capital for the area of the base. So to find the area of the the square, it's going to be the length times the width or simply B^2. So the area of the base is B^2. So therefore the volume of this entire square base pyramid is 1/3 time the side length squared multiplied by the height of the pyramid. So it's going to be 1/3 6 ^ 2 * 10. 6 * 6 is 36 and 1/3 of 36 or 36 / 3 that's 12 and 12 * 10 is 120. So the volume is 120 cubic units. So let's say if this was 6 in, 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 8 in and the height let's say it's 12 in. So feel free to pause the video and find the volume of this pyramid. So the volume is going to be 1/3 time the area of the base multiplied by the height of the pyramid. And the area of the base as we defined in the last example is going to be the side length of the square. So basically just ^ square. So now example lowerase is 8 and the height is 12. So it's 1/3 * 8^ 2 * 12/3 of 12 or 12 / 3 that's 4 and 8^ 2 is 64. 4 * 64 is 256. So the volume is 256 in. And so that's the answer for this example. Now let's talk about finding the slant height of squarebased pyramid. So let me just draw this first. want to draw this carefully. Now the red line is the height of the pyramid as we talked about before. So this makes right angle with the 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 which represents the length of the green dash line and the red dash line that represents the height of the pyramid. We're going to say represents the side length of the square. Now we've talked about the volume of pyramid. It's 1/3 ^2 * or 1/3 the area of the base * But now if you want to find 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^ squ. 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 the triangle on the right side. You have triangle on the left, one in the back, and one in the front. So that triangle, I'm just going to redraw just 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 Now whenever you have triangle if you know the base and if you know the height of the triangle to find the area is simply 12 base time height. But in this case the height is the slant height. So the area becomes 12 base time slant height which is 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 it 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 * the area of each of those triangles, which is 12 * So 4 * half is 2. So therefore, the surface area, you can find it using this equation. B^2 + 4 mean 2 BL. So that's the formula I'm going to use to find the surface area of 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 cm and you're given the length of the base. Let's say it's 8 centimeters. It's 8 by 8. Go ahead and calculate the surface area of this object or figure and also find the lateral area as well. Now the first thing would do is find the lateral area. The lateral area is simply 2 As you mentioned before, it's 12 base times height or in this case height. 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, in our example is 8 and is 15. So, it's 2 * 8 * 15. Now, 2 * 15 is 30. And 30 * 8. If 3 * 8 is 24, 30 * 8 is 240. So it's 240 square cm. 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 ^2 + 2 BL. B^2 that's going to be H2 and 2 BL is 240. 8^2 is 64. 64 + 240 that's going to be 304. So the surface area is 304 square cm. And so now you know how to find the surface area and the lateral area of square base pyramid. So these are the two equations that you need. Now let's work on another problem. Let's say the height is 12 in. So that's going to be the height of the pyramid. And let's say the base is 10 in long and 10 in 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 minute and work on this problem. Feel free to pause the video too. So let's start with the volume. It's 1/3 base time height, which is 1/3 the side length squar time the height of the pyramid. is 10 and the height is 12. So 1/3 of 12 is 4 and 10^ 2 is 100. So in this problem, the volume is 400 in. 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 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 this height. But what is the base of the triangle? Well, if 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 five. 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 is 10, 5 is / 2. It's half of the base length. So to find for the square base pyramid, L^2 is equal to / 2^ 2 + H^2. So you can use that formula if you ever need to find slant height, if you know the length of the base and the height of the pyramid. So over 2 is 5 and is 12. 5^2 is 25. 12^2 is 144. When added together that will give you 169. So now we got to 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 BL. So it's 2 * 10 * 13. Now 10 * 13 is 130 and 2 * 130 is 260. So that's the lateral area. It's 260 square in. 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 ^ 2 + 2 So ^ 2 that's going to be 10^ 2 + 2 * 10 * 13. So 10^ 2 is 100. And we know that 2 * 10 * 13 that's the lateral area which is 260. giving us total surface area of 360 square in. 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 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 nice big picture so you can clearly see everything. And this is right angle. And in red, just like 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 square base pyramid. But here, this is 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 1/3 time 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 12 base times height. So make sure you distinguish the height of the pyramid which I'm using capital versus the height of the triangular base which I'm going to use as lowerase 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 five. 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. Five 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 1/3* the area of the triangle which is 12 base time height multiplied by the height of the pyramid. So the base of the triangle is 5 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. 12 of 12 is six. So now we don't have to worry about those two numbers anymore. 1/3 of 15 or 15 / 3, that's 5. And we still have another five to deal with. Now 5 * 6 is 30. And 30 * 5 is 150. So the volume is 150 cubic units. So now you know how to find the volume of triangular pyramid. Now let's focus on finding the surface area of triangular pyramid. So let's draw picture first. And let's say the line in green is the slant height. So we have the slant height 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 typical prejudge course. So, if it's equilateral triangle, then instead of having height, this is also going to be It's going to be base length. And this two would be 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 3 over 4 * ^ 2 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 triangle in the front. Now, sometimes they may give you the area of the base. If that's the case, you could just replace with that number. But if you need to find the area of an equilateral triangle, you can use this formula. Now there's three lateral faces and the area of each of those faces is 12 base times height. But we need to use the slant height instead. So it's 12 * 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 12 base * height. So depending on what you're given, sometimes you could use this formula. 12 base * height plus 3 over 2 * if you combine 3 * 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 length of the red line, we're going to say it's 10.4 in. 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 in long. Using this information, 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 lateral 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. 3 b^2 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 3 * 12 base time height. Now the base is 12. The height of the triangular base that's 10.4. 24 is 15. So now half of 12 is 6 and then 6 * 10.4 that's 62.4. So that's the area of the base. Now to find the lateral area, it's basically 32 * So 3 * 12, that's 36 / 2, that's 18 * 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 in. 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 is 12. ^2 12^2 is 144 / 4 that's 36. So this is going to be 36 3. 36 roo3 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 rounded value. But the answer for this problem is 332.4. That's how you could find the surface area using that equation. Let's try one more problem. So let's say if we're given slant height of and we're given base length let's Say it's 8 and the 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 3 * 12 base * the slant height. So the length of the base is 8 and the slant height is 20. Now half of 8 is 4 and 4 * 20 is 80 and 3 * 80 is 240. So the lateral area is 240 square cm. 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 3 over 4 B^2. So is 8 and the lateral area is 240. 8^2 is 64 and 64 / 4 is 16. So right now we have this the surface area is 16 3 + 240. So that's the exact answer. But if you want to get decimal value multiply 16 by the of 3 and that should give you 27.7 if you round it to the nearest 10th and then let's add 240 to it. So the surface area is 267.7 square cm. So that's the answer.