Surface Area of a Square 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 square pyramid and remember surface area is the total area of the outside or surface of a three-dimensional shape when it comes to square pyramids we have five total faces the base and then 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 now there are a couple of different ways to work through these so i'll show you some different options as far as finding the surface area of a square pyramid so let's jump into our example where we have a square pyramid with a slant height of 5 inches and then the sides of the base are 6 inches the first thing that i'm going to do is write out surface area equals the area of the base plus the lateral area that's going to get us started then we'll plug in our information let's use the net on the right 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 so we're going to start with the area of the base now for this square pyramid the base is six inches by six inches and we can see on our net we have six inches and then all of the other sides are six inches as well so to calculate the area of a square we can do i'm going to come to the right here and show some work one of the sides squared and we know that we have six inch sides so let's plug in six and six squared means six times six so the area of the base is 36 square inches so let's put that in to our net here and now we'll find the area of all of our lateral faces and since this is a square pyramid that means it's a regular pyramid meaning the base's sides are all the same so we just need to find the area of one of the triangular lateral faces and we will have the area of all of them because they are all the same so we can use the formula area equals one-half times base times height so area equals one-half times base times height now you can also use base times height divided by two multiplying by half or dividing by two same thing whatever formula you like to use for the area of a triangle now we plug in so area equals one-half well the base of this triangle is six inches so let's plug in six times the height of 5 inches so 6 times 5 is 30 so we end up with one half times 30 and we will squeeze this in so the area is 15 square inches for each of our lateral faces so let's put that in here 15 square inches for each of these now that we have the area of each face we can add everything to get the total surface area so let's plug it all in and calculate here so the area of the base was 36 square inches so let's put in 36 for the area of our base plus we need the lateral area so all of the triangular lateral faces so each had an area of 15 square inches so we need four 15's here and then we'll add everything to get our total surface area so we have 4 15s so 15 times 4 or 15 plus 15 plus 15 plus 15 either way it's going to give us 60. so let's break this down and we'll end up with 36 plus 60 and again that 60 is coming from those four 15s so i added those four 15s together first or you can do 15 times four and to wrap it up 36 plus 60 is going to give us a total surface area of 96 square inches and that's our final answer now i would like to show you one other way we can solve to find the surface area of a square pyramid we can use the formula surface area equals the area of the base plus one half p and then s where the p stands for the perimeter of the base and the s stands for the slant height now i do want to mention that an l can be used to represent slant height as well you'll see both but it doesn't matter which they mean the same thing now that we have our formula written out we can plug in so surface area equals well the area of the base we already calculated that it was 36 plus one-half times the perimeter of the base we need to do six plus six plus six plus six or six times four so the perimeter of our base is 24 times the slant height of five inches right here once we have everything plugged in we can solve and we can start simplifying or plug this in to a calculator and solve it in one step i'm going to simplify here i'm running out of room but we can make it work so surface area equals 36 plus one half times 24 times five well one half times 24 is 12 times 5 gives us 60. so again i did one half times 24 that gives us 12 times 5 is 60. now we end up with 36 plus 60 and that's going to give us a total surface area of 96 square inches so we got the same answer as we did up top so two different ways there to find the surface area of a square pyramid we can find the area of each face and add all of those areas together or we can use the formula right here surface area equals the area of the base plus one-half times the perimeter of the base times the slant height now the purpose of that formula is to condense everything into one formula rather than adding all of those faces up so you can see that it's kind of similar because we have the capital b the area of the base and then this portion right here this gives you the lateral area so it will take all of those faces the areas and find the total area of those faces in one step there so again the purpose of that formula is to condense everything into a formula without having to add all of the faces both methods work so whatever works best for you go with that so there you have it there's how you find the surface area of a square pyramid i hope that helped thanks so much for watching until next time peace