Surface Area of a PyramidDate: 09/01/97 at 19:25:10 From: Daisy Villagomez Subject: Surface Area of a Pyramid Hello. I would just like to know how to get the surface area of a pyramid. Thanks. Date: 09/20/97 at 14:21:41 From: Doctor Sonya Subject: Re: Surface Area of a Pyramid Hi Daisy! You didn't say what kind of pyramid, but I'm guessing you want the surface area of a regular square pyramid. That's a pyramid with a square base and four sides that are all the same. Can you imagine what such a pyramid looks like? If you have some toothpicks and clay, you can try building one by making four toothpicks into a square (held together at the corners with clay) and then attaching one toothpick to each corner and bringing all four of them together at the top. You can even use mini-marshmallows to hold the toothpicks together if you want. That's how I build my geometry models. Just in case you don't want to build a model, the pyramid has a square base and four equal triangular sides. Do you see why the side are triangles? To get the surface area of the pyramid, you need to find the area of the base and the area of each of the sides, and then add them up. Whenever you talk about pyramids, there are two things you have to know: the length of the sides of the base and the height. These two things should be given in your problem. The height of a pyramid is how tall it is. If you were to build a hollow pyramid, tie a string to the top, and then let it down into the middle of the pyramid until it hit the floor, the length of the string would be the height. This length is also called an altitude, because it is perpendicular to the floor. Now that we know exactly what we are talking about, let's assign some values to our lengths. We'll make them variables becasue you want a general formula. Call the length of one side of the base "b" and the height of the pyramid "h". The area of the base is easy to find. What's the area of a square with side length b? Now we need to find the areas of the triangles on the sides. Remember that the area of a triangle is: (1/2)(base)(height) The triangles have base b, but what is their height? We can use the Pythagorean Theorem to find the height of the triangles. Picture a triangle inside of the pyramid with one side straight "down" through the pyramid from the very top to the center of the base, another side from the center of the base to the midpoint of one side of the base, and the third side from the midpoint of one side of the base to the top of the pyramid. Drawing a picture (or using more marshmallows) might help you visualize this. This third side is the height of the triangles we are looking for. Fortunately, we know the other two sides: h and b/2. Think about how we got these. These are also two legs of a right triangle. Remember that the Pythagorean Theorem says that if the two legs of a right triangle have length A and B, the the length of the hypotenuse, C, can be found with the equation: A^2 + B^2 = C^2. (A^2 means "A squared") The two legs of our right triangle above are of lengths h and b/2, so (length of hypotenuse)^2 = h^2 + (b/2)^2. If we solve this equation for the length of the hypotenuse, we find that it is: SQRT(h^2 + (b/2)^2). So the area of one triangular face is: (1/2)(base)(height) = (1/2)(b)(SQRT(h^2 + (b/2)^2). That means the surface area of the entire pyramid is the areas of the four triangular faces (remember they're all the same) plus the area of the base. You can use this same technique to find surface areas of pyramids with other bases. Let us know if you need some extra help! -Doctors Ziggy and Sonya, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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