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 a 2d shape like a square is all of the space inside of the shape right in other words you can take a pencil and color in the area of the square which we've learned we can calculate by multiplying base times height now if we look at a 3d object like a 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 a pencil and color in one of the faces of the cube and still calculate the area by multiplying base times 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 inches 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 a triangle is one-half base times height so let's do a mini area problem we have the base which is six inches but we don't have the height directly so we have to use a little a squared plus b squared equals c squared action to find the height first plugging in all the numbers we're trying to find a b equals three since this is half of six and c is 6. solving for a 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 one half base times height and we'll get roughly 15.6 inches 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 inches 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 a rectangle and we can find the area of this one face as base times height so 6 times 10.5 which is 63 inches squared now this is only the area of one face and we have a total of one two three identical faces remember there's one in the back taking 63 times three faces will get a surface area of 189 inches 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 inches squared that's it that's our answer next problem we have a cylinder and it looks a 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 a can of soup is a cylinder a roll of paper towels is a cylinder and if you look at it from the top the bases are circles to find the area of the base the formula for the area of a circle is pi r squared from the diagram we're getting that the diameter is 2 feet so we know the radius is one foot because the radius is just one half of the diameter plugging into the formula we'll get about 3.14 feet squared for the area of one base let's double it to find the area of both bases and we'll get 6.28 feet squared our next step is a little different than what we've been doing because it's a 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 a paper towel roll that's a cylinder let's think of the outside face as one of the paper towels if you unravel one sheet of a paper towel it's just a rectangle right for the dimensions we know that the length of the paper towel is seven feet 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 r and we found the radius earlier so the height is 6.28 feet 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 times height so 7 times 6.28 and we're going to get an area of 44 feet squared for our final step we can add up the areas 6.3 plus 44 and get 50.3 feet squared that's it last challenge problem this is a hexagonal hex hexagonal prism well the shape of the base is a hexagon where all six sides are 10 yards and this little length from the center to one of the sides is four yards if i redraw it from the top it looks like this and i don't know about you but i've never memorized the formula of a hexagon although apparently there is one if you think about it a 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 a triangle is one-half base times height so one-half ten times four this equals 20. then because there's six of these little triangles in a 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 two and we'll get an area of 240 yards squared for both of our bases now finding the area of the outside faces if we just look at one face it's a rectangle right so base times height which is 10 times 80 is 800 yards squared and there are a total of six faces so 800 times 6 equals 4 800 yards 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 240 plus 4800 is 5040 yards squared which is our final answer alright those were all of the problems i wanted to go over today if this video helped you out be sure to give it a thumbs up and subscribe for more math tutorials like this i'll see you guys in the next video