Rectangular Prism Volume Surface Area and Diagonal Length Rectangles Geometry

in this video we're going to talk about how to calculate the volume the surface area and the diagonal length of rectangle so let's try this problem what is the volume of rectangle that has length width and height of 12 inches 6 inches and 5 inches respectively so let's draw picture so here is rectangular prism and it has length of 12 inches width of 6 inches and height of 5. so how can we find the volume of this prism the volume of any prism is basically the area of the base multiplied by the height the height of the prism is 5. the length is 12 and the width is 6. so we already have the height in this equation we don't have to worry about it but what is the area of the base in this particular picture this is the base of the rectangular solid based on the way strong and the area of the base the area of that blue portion is basically the left times the width so we can replace with times so the volume of rectangular prism is simply the length times the width times the height the length is 12 inches the width is six inches and the height is five inches twelve times five is sixty and sixty times six well six times six is thirty six sixty times 6 is 360. you just have to add the extra zero now what about the units so what are the units of volume in this problem inches times inches times inches that's equal to inches cube or cubic inches so that's the volume in this example it's 360 cubic inches number two what is the surface area of rectangular prism with the dimensions 15 centimeters by eight centimeters by nine centimeters so let's say that this is the length this is the width and this is the height well let's begin by drawing picture just like we did before doesn't have to be perfect just simple rough sketch my prism looks bigger in the back but you can work with it so let's say this is the length width and the height how can we come up with an equation that was that's going to help us to to find surface area of this rectangular prism the surface area is basically the area of all six faces of this prism or this rectangle so the area of the front is the same as the area of the back face or the just the back side of the rectangle so that's highlighted in blue notice that it's the width times the height same thing on this side this is the width and this portion is the height so the area of the front and the back is just times and because there's two of them it's going to be 2 wh now the six sides we need to cover so far we've covered the front and back so just two out of six now the next thing is you have the area of the bottom surface and the area of the top surface so the bottom surface is units long and it's units same thing for the top so the area is times but because we have the top and the bottom we got to multiply by two so it's going to be two times now the last thing we need to worry about is the right side and the left side the right side has length of but height of so the area is lh but we've got to multiply by 2. so this is the formula that you need to calculate the surface area of rectangular prism now let's go ahead and plug in everything that we have so the width is eight centimeters and the height is nine centimeters now the length is 15 centimeters and the width is eight centimeters and then two times lh that's gonna be two times fifteen times nine so now we just gotta do some math eight times nine that's 72 and two times 72 is 144. now 2 times 15 is 30 and what's 30 times 8 well 3 times 8 is 24 if you add the zero that's going to be 240. and here we have 2 times 15 again that's 30 30 times 9 is 270. so now we just got to add so what's 240 plus 270 well 200 plus 200 is 400 40 plus 70 is 110 so 400 plus 110 that's going to be the 510 now 500 plus 144 that's 644. if you add 10 to it that's going to be 654. so that's the surface area and whenever you're dealing with area the units is going to be unit squared so it's going to be square centimeters or centimeters squared so that is the surface area of this figure 654 square centimeters so now you know how to find it number three what is the length of the diagonal of rectangular prism with the dimensions 14 centimeters by nine by seven so once again this is the left the width and the height so let's start with picture so we need to find the distance between one edge of the rectangular prism to the other edge so basically we're looking for the length of the yellow line so how can we go ahead and find that how can we find the length of that line well before we come up before we write the equation let's talk about how to get it so you could understand why the equation is the way it is so notice that you could form right triangle so i'm going to draw another diagonal that remains on the bottom surface and notice the triangle that forms this is right angle so i'm going to call the length of the diagonal which we're trying to find the yellow line let's call it we want to find the distance between these two points and this side is the height of the prism that's and the left for the red diagonal that is on the bottom surface let's call this so we have this triangle the hypotenuse is which is what we're trying to find is the height and is the red diagonal that's on the bottom face of the rectangular prism now for any right triangle we have the pythagorean theorem squared is equal to squared plus squared so in this case squared is equal to squared plus squared now notice that there's another diagonal that we could focus on i'm going to highlight it in green so this is as we mentioned before but notice that we could form another right triangle where the hypotenuse is right here so this is and this is so for that second triangle which looks like this we have hypotenuse of we have the width and the length so therefore using the pythagorean theorem squared is equal to squared plus squared now if we combine these two equations if we replace squared with what it's equal to squared plus squared we're going to have this equation which is what we want squared is equal to squared plus squared plus squared and now let's take the square root of both sides so here's the final equation is equal to the square root of squared plus squared plus squared so if you have the left the width and the height you can find the diagonal length using that excuse me using that equation which comes from the pythagorean theorem so now let's go ahead and find the answer so it was 14 by nine by seven so those are the dimensions of this particular rectangular prism and this is the length with and the height so let's go ahead and use this formula to calculate the length of the diagonal so is fourteen is nine is seven fourteen squared is one ninety six nine squared is eighty-one and seven squared is 49. so if we add those three numbers we should get 326 if typed in everything correctly and the square root of 326 as decimal is 18.055 and the unit is going to be centimeters so make sure you get an answer that is greater than each of these numbers individually the length of the diagonal has to be longer than the longest side of the rectangle it's just the way the math works so this is the answer so now you know how to find the volume the surface area and the length of the diagonal using these equations so that's all you need to know for rectangular prisms so thanks for watching this video and have great day you