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How to Find the Volume and Surface Area of a Pyramid
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The volume is defined as one third the area of the base times the
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How to Find the Volume and Surface Area of a Pyramid
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Hi, I am Eric Stone from South Burlington High School in South Burlington, Vermont, here for About.com. Today, we are going to try to find the volume and surface area of a rectangular pyramid -- kind of like the pyramids at Giza. In fact, I will show you how to find the total volume and the surface area of the Great Pyramid at Giza!Lets start with the volume. The volume is defined as one third the area of the base times the height. So if I want to know the volume of my pyramid, I not only need to know the lengths at the bottom, but I also need to know how tall the pyramid is -- we will call that h. Since it is a square base, the length of any one side squared will be my b and then my height will be h. V equals one third L squared times h.
Well the pyramid at Giza has the following dimensions. It is 755.5 feet wide and it is 450 feet high. Finding the volume: straightforward. The volume is equal to one third the length, the width, the height: the area of the base times the height. Since the base is 755.5 by 755.5 we can plug that in and the height of the pyramid is 450 feet. So if I multiply one third times the length times the width times the height, I get 85,617,037.5 cubic feet. That is a lot!The surface area is a little more tricky. Because you have to see that there is a lot that you have to add together. There is the square base that we started with -- we need to find the area of that. But also, we need to be able to find the areas of each of the other sides: this triangle, this triangle, the triangle in the back and the triangle on the side. There are four of them.
And finding that area can be a little tricky. The reason for that is, the formula for the area of a triangle is equal to one half base times height -- where the base is simply the length from here to here. But the height is traveling along the length of that pyramid. See, it is not the height of the pyramid, it is this distance, it is the angular distance along the face.
So lets go back and talk about the pyramid at Giza. If I want to find the surface area, lets recap. I need to find the area of one triangle. So I use my pythagorean theorem to find out the length of the side of the pyramid. Well, a-squared plus b-squared equals c-squared. We said that 450 feet was the height of the pyramid and we said that 755.5 feet was the length of the base -- and we need to use half of that, because we only want one half of that side. Do the math and you end up with 587 feet.
So, to find the area of the triangle, it is going to be one half times the length of the base times the height of the triangle.
If I want to find the total area of the Great Pyramid at Giza, I want to take the area of the four triangles, plus the base. Four triangles at 221,592.5 square feet plus 570,780.25 square feet, which is the area of the base -- add them up and we get 1,457,150 square feet for the area of the Great Pyramid at Giza.
Thanks for watching! To learn more, visit us on the web at About.com.
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