Surface Area of a Rectangular Pyramid | Math with Mr. J

[Music] welcome to math with mr j [Music] in this video i'm going to cover how to find the surface area of a rectangular pyramid and remember surface area is the total area of the outside or surface of a three-dimensional shape when it comes to rectangular pyramids we have five total faces the base and four lateral faces the lateral faces are the triangular faces going around the pyramid so to speak we need to find the area of all of these and then add them together to get the total surface area so let's jump into our example and start by writing out surface area equals the area of the base plus the lateral area then we need to plug in our information so we need to calculate the area of the base and all of the lateral faces then add them together let's use the net on the right side of the screen to help us do this you can think of a net as an unfolded 3d shape it shows us all of the parts so to speak let's start with the area of the base so we have a rectangular base so we can use the formula area equals length times width now i'm going to use the bottom portion of the screen to show all my work as far as where i'm going to calculate the areas so let's come to the bottom left area equals length times width now we plug in so our base has a length of 9 meters and a width of 5 meters so 9 times 5. now if you use 5 for the length and 9 for the width that would work as well don't get too held up on which one is length or width because 9 times 5 is the same as 5 times 9 both will give you the correct area so the area of our rectangular base is 45 square meters i'm going to put a capital b above that so we know that that is the area of the base now that we have the area of the base we can move to the lateral faces and since we do not have a regular base regular means all of the sides are the same and this is not the case this means that we will have two different slant heights and therefore two different areas for our lateral faces these will have the same area the top and bottom of the net because they have the same base 9 meters and same height 8 and 38 hundredths meters the left and right will have the same area because they have the same base of 5 meters and same height of 9 and 18 hundredths meters so let's calculate the top and bottom lateral faces of the net first these would be the front and back lateral faces of the 3d shape on the left hand side of the screen now since we're working with triangles we can use the formula area equals one-half times the base times the height so area equals one-half times the base times the height now you can also use base times height divided by two for triangles multiplying by a half or dividing by two they will get you the same answer so whatever formula works best for you that's what you can use for finding the area of these triangles now we plug in so again we're going to do the top and bottom from the net first so one half times the base well these have a base of nine meters so let's plug in nine times the height of eight and thirty eight hundredths and now we're ready to calculate so the area of those lateral faces is going to be 37 and 71 hundredths and this is square meters now i'm going to label this with one star above it so we know that it is the top and bottom lateral faces of that net so we can move to the left and right lateral faces now and we have triangles so we will write out our formula area equals one-half times base times height and now we plug in so one half and these lateral faces have bases of five meters so let's plug in five and a height of nine and eighteen hundredths meters so nine and eighteen hundredths now we calculate so that's going to give us an area of 22 and 95 hundredths square meters and i'm going to label this with two stars so we know which lateral faces have the area of 22 and 95 hundredths square meters so once we have all of the areas calculated for our faces we can fill in our net to show exactly which areas go where and it will help us stay organized so we know that the area of the base was 45 square meters so let's plug that in there and then our one star lateral faces 37 and 71 hundredths square meters so let's label those and i'll put this one on the outside here so 37 and 71 hundredths square meters and this goes right here and now we have the two star lateral faces the left and the right 22 and 95 hundredths so let's squeeze this in here 22 and 95 hundredths square meters and then 22 and 95 hundredths square meters goes right here so we again have the areas for all of our faces the base and then the lateral faces so now it's time to add everything up to calculate that total surface area now let's come back up top and add these so we have surface area equals the area of the base 45 plus now we have all of our lateral faces let's start with the top and bottom so 37 and 71 hundredths plus 37 and 71 hundredths now i'm going to have to go below here in order to squeeze everything in and now we have the left and right so 22 and 95 hundredths plus 22 and 95 hundredths so you can see that we added five numbers together five areas those represent our five faces now it's time to calculate so the total surface area equals 166 and 32 hundredths square meters and that is our final answer so there you have it there's how you find the surface area of a rectangular pyramid i hope that helped thanks so much for watching until next time peace you