FIND THE SURFACE AREA OF PRISMS AND CYLINDERS GEOMETRY LESSON

Hey guys, today we're going over how to calculate the surface area of prisms and cylinders like these. Before we get into the problems, let's first start off with what does surface area even mean? Well, the area of 2D shape like square is all of the space inside of the shape, right? In other words, you can take pencil and color in the area of the square, which we've learned we can calculate by multiplying base time height. Now, if we look at 3D object like cube, the surface area is all of the space on the outside of this 3D shape. In the same way we calculated area, we can take pencil and color in one of the faces of the cube and still calculate the area by multiplying base time height. But notice how we have all of these other faces that we haven't colored yet. For example, if you think of dice or an ice cube, they have six total faces or sides. Meaning, we need to calculate the individual areas of all of the faces and add it all up. That is the surface area. For our next problem, we're going to do the same exact thing. We're going to first identify the shape of the base. So, you know how in the last problem the bases were rectangles? Now we have, you guessed it, triangles for bases. And these red lines indicate that each side length on the triangles are the same length. If it's hard to visualize, this base looks just like this. If you view the prism from the top, and each side length is 6 in. Our next step is to find the area of the bases, which basically means find the area of these triangles. If you remember the formula for the area of triangle is 12 base time height. So let's do mini area problem. We have the base which is 6 in. But we don't have the height directly. So we have to use little a^2 + b^2= c^² action to find the height first. Plugging in all the numbers, we're trying to find b= 3 since this is half of 6 and is 6. Solving for we'll get that the height is the square root of 27. Finally, we can calculate what we initially set out to do, which is to find the area of the base. Let's plug everything into the formula 1/2 base time height and we'll get roughly 15.6 in squared. This is just the area of one base. And since there are two bases, let's double that. So the area of both bases is about 31.2 in squared. Next, let's find the area of the other faces on the prism. And these are usually just rectangles. If we look at this face, it's rectangle. And we can find the area of this one face as base time height. So 6 * 10.5, which is 63 in squared. Now, this is only the area of one face, and we have total of 1, two, three identical faces. Remember, there's one in the back. Taking 63 * 3 faces, we'll get surface area of89 in squared for all of the faces on the side. Finally, we've done all the hard stuff. Let's add them together to get the total surface area. We'll get 220.2 in squared. That's it. That's our answer. Next problem. We have cylinder and it looks little different than your typical prism, but the same steps still apply. Looking at the shape of the base, the bases are just circles. If you think of it, can of soup is cylinder. roll of paper towels is cylinder. And if you look at it from the top, the bases are circles. To find the area of the base, the area of circle is From the diagram, we're getting that diameter 2 ft. So, we know the radius is 1 ft because the radius is just 1/2 of the diameter. Plugging into the formula, we'll get about 3.14t squared for the area of one base. Let's double it to find the area of both bases and we'll get 6.28 ft squared. Our next step is little different than what we've been doing. Because it's cylinder, it doesn't have all of these little individual faces for us to calculate and add up. Instead, it's like this one continuous face. If we think about paper towel roll, that's cylinder. Let's think of the outside face as one of the paper towels. If you unravel one sheet of paper towel, it's just rectangle, right? For the dimensions, we know that the length of the paper towel is 7 ft. That doesn't change. And the height of the paper towel is the circumference of the circle. The formula for the circumference is 2 pi and we found the radius earlier. So the height is 6.28 ft. Going back to what we originally set out to do, which is find the area of this paper towel rectangle, we're going to multiply the base time height. So 7 * 6.28 and we're going to get an area of 44t squared. For our final step, we can add up the areas 6.3 + 44 and get 50.3 ft squared. That's it. Last challenge problem. This is hexagonal hex hexagonal prism. well, the shape of the base is hexagon where all six sides are 10 yards. And this little length from the center to one of the sides is 4 yards. If redraw it from the top, it looks like this. And don't know about you, but I've never memorized the formula of hexagon, although apparently there is one. If you think about it, hexagon is just six little triangles put together. So if we find the area of one of these triangles, we can multiply it by six to get the area of the entire hexagon. Okay, the area of triangle is 1/2 base time height. So 12 * 4 this equals 20. Then because there's six of these little triangles in hexagon, let's multiply it by six and we'll get the total area of this one hexagon is 120 yards squared. But because there are two bases, one at the top, one at the bottom, let's multiply by 2 and we'll get an area of 240 yard squared for both of our bases. Now finding the area of the outside faces. If we just look at one face, it's rectangle, right? So base times height, which is 10 * 80 is 800 yard squared. And there are total of six faces. So 800 * 6 = 4,800 yd squared. This is the area of all of the outside faces on the side. In our final step of this problem and the final step in this video, let's add up the areas of the bases and faces. plus 4,800 is 5,040 yard squared, which is our final answer. All right, those were all of the problems wanted to go over today. If this video helped you out, be sure to give it thumbs up and subscribe for more math tutorials like this. I'll see you guys in the next video.