12 2 Surface Areas of Prisms and Cylinders

section 122 is on surface areas of prisms and cylinders prism is polyhedrin and polyhedrin is three-dimensional shape that has two parallel congruent faces that are called the bases the faces that are not the bases are called lateral faces and they intersect at lateral edges and the distance between the bases is called the height let's take look at an example of one here's hexagonal prism okay it has hexagon on the top and hexagon on the bottom those are the bases because they are parallel and congruent to each other the lateral faces are the other faces that connect the two hexagons and those are shaped as rectangles the height is represented by this blue dashed line in the middle and that's the distance between the bases and the lateral edges are these edges right here where the lateral faces meet each other the lateral area of prism is the sum of all of the areas of the lateral faces and the surface area is the lateral area plus the area of the bases so in this case the lat lateral area is just the area kind of around the outside the surface area is the area of all of the exposed surfaces here are some formulas the lateral area of prism is represented by the formula Lal pH where is the perimeter of the base and is the height the surface area of prism is represented by the formula = + 2B or since is the lateral area you could say = pH + 2B again where is the perimeter of the base is the height of the prism and capital is the area of the base Let's do an example here find the lateral area and the surface area of this triangular prism so what I'd like to do first of all is identify what the base is now you might think the base is on the bottom in that case it would be this 12 by 8 rectangle but that's not the base base doesn't mean bottom base when you're talking about prism is one of two parallel congruent faces and in this case the base is right triangle it's this right triangle in the front here maybe I'll just go ahead and draw that that is 12 by 12 and this is labeled so there's couple things need to know about that need to know the perimeter of that and I'll need to eventually find the area of that so let's start by finding the perimeter don't know the length of the hypotenuse here so I'll use the Pythagorean theorem 122 + 122 = c^2 and working that all out is approximately equal to 1697 so that means that the perimeter of this base is going to be 12 + 12 + 16. 97 or approximately 40.97 40.97 okay now the height of this prism is the distance between the bases and the height is going to be represented by this 8 right here the triangle in the front is 8 away from the triangle in the back so to find the lateral area is perimeter time the height or 40.97 * 8 or approximately squared okay so that's the area of all of the rectangular faces so the rectangle on the bottom the rectangle on the left and the slanty rectangle that's kind of looks like the hypotenuse now to find the total surface area what need to add to the lateral area is the area of the bases so to find the area of the base is going to be the area of this triangle right here and for that I'm going to use the triangle formula 12 little * now here's where it gets little confusing big stands for the area of the Triangular base this little little stands for the linear base of the right triangle so big is equal to2 12 the base of that right triangle is 12 the height of that right triangle is 12 or the area of the base is 72 now to find the total surface area I'm going to do the lateral area which is 32776 + 2 * the area of the base which gives me approximately 47176 Square millim so the trickiest part think of prism is figuring out what the base is what shape is the base and remember the base is not always on going to be on on the bottom cylinder is solid that has circular bases connected by curved surface when see cylinder tend to think of soup can the bases are circles top and bottom and the lateral surface if you were to like unroll this is going to be in the shape of rectangle if you've ever taken the label off of soup can you can see that the shape is rectangle so the formula for the lateral area of cylinder is represented by = 2i where is the radius of the circular base and is the height the distance between the two circles the total surface area of the cylinder is going to is represented by = which is the lateral area plus 2 * the base but because the lateral area is 2 pi RH and know the area of the base is going to be the area of circle it's easier to use this formula here = 2i RH that's the lateral area plus 2K 2 those are the areas of the two circles the top and the bottom so let's take look at an example here find the lateral area and surface area of this cylinder so we're given that the diameter is 15 so that means that my radius is going to be 7.5 cuz the radius is half of the diameter and my height is 18 so from here it's just very straightforward fill these into the formula so the lateral area is 2 pi * * and that's going to give me approximately 23 squared so that's the area of the rectangular label on the soup can my total surface area is going to take my lateral area 8482330064 me approximately 1, 12166 squared and here's our last example believe yes this is our last example find the diameter of the base of cylinder if its surface area is 464 Pi square cm and its height is 21 cm so know the total surface area so I'm going to write the formula out here = 2i RH + 2 piir 2 okay know the surface area is 464 pi and know my height is 21 what need to find is the diameter but think I'll start by finding the radius and then can double that to get the diameter so let's plug in the numbers we know surface area is 464 pi and that equals 2 pi don't know is 21 + 2 piun r² let's clean that up little bit 464 pi equal 2 * 21 or 42 pi + 2 pi 2 now remember what I'm trying to find is so is kind of like my notice that this is quadratic equation because have an an and term with no so let's see if we can reduce the numbers little bit here I'm going to divide everything by 2 pi nice thing about that is all of my pies are going to cancel out and that's going to leave me with oops 232 = 21r + 2 my Pi's cancel out and my twos cancel out leaving me with 2 my Pi's cancel out 42 over 2 is 21 my Pi's cancel out and 464 over 2 is 232 so this is quadratic equation let's make it equal zero and write it in more of standard form okay and so have r^2 + 21 - 232 = 0 I'm going to use the quadratic formula in my calculator and get that = 8 and = -29 and of course the radius can't be negative number so it can't be -29 so if my radi is 8 my diameter is twice that so my diameter is 16 and my units were CM so in this section we looked at the lateral area and the total surface area of prisms and cylinders just make sure you don't have to have the formulas memorized but make sure you're familiar with them and know how to plug things in