Finding Surface AreasDate: 06/17/2002 at 18:35:28 From: Megz Subject: Measurement. I don't understand the formulas to figure out surface area on the different kinds of figures. It is really confusing to me, and hard to explain about what I don't understand. It's basically the whole thing about it. Megz Date: 06/17/2002 at 21:19:27 From: Doctor Ian Subject: Re: Measurement. Hi Megz, The basic idea is that to find the surface area of a figure, you break the figure up into individual sides or 'faces'; find the area of each face; and add them all up. Sometimes this results in a very compact formula that doesn't look very much like you did that; but it's still what's going on. Let's look at a couple of examples. How about a cube? There are six faces to a cube. (If you forget this, recall that the sides of dice are numbered 1 to 6.) Each face of a cube is a square, and the length of each side of the square is the same as the length of an edge of the cube. The area of a square is the length of a side multiplied by itself. So the surface area of a cube is surface area = area of face 1 + area of face 2 + area of face 3 + area of face 4 + area of face 5 + area of face 6 = edge*edge + edge*edge + edge*edge + edge*edge + edge*edge + edge*edge = 6 * edge * edge = 6 * edge^2 Now, suppose we have, not a cube, but a rectangular prism (like the shape of a cereal box). We still have six sides, but now they come in pairs, and each side is a rectangle. Two of the rectangles have dimension width * height; two have dimension width * length; and the remaining two have dimension length * height. So the surface area is surface area = area of face 1 + area of face 2 + area of face 3 + area of face 4 + area of face 5 + area of face 6 = width*height + width*height + width*length + width*length + length*height + length*height = 2 * (width*height + width+length + length*height) Let's look at one more example: A cylinder. There are three 'faces' to a cylinder: a circular one at each end, and the big curved side. The area of each circle is pi times the square of the radius. So the surface area is surface area = area of circle 1 + area of circle 2 + area of side = pi * radius^2 + pi * radius^2 + area of side What about the side? Well, imagine that you make a cut from top to bottom, and unroll the side. You get a rectangle, right? The height of the rectangle is the height of the whole cylinder. What is the width of the rectangle? It's the circumference of the circles! So we can complete the formula: = pi * radius^2 + pi * radius^2 + height * circumference = pi * radius^2 + pi * radius^2 + height * pi * diameter = pi * radius^2 + pi * radius^2 + height * pi * 2 * radius Now, each of these terms has pi in it, so we can factor that out: = pi * ( radius^2 + radius^2 + height * 2 * radius) Each term in parentheses also has a radius in it, so we can factor that out too: = pi * radius * ( radius + radius + 2 * height) We can add the radii together: = pi * radius * (2 * radius + 2 * height) And now we can factor out a 2: = 2 * pi * radius * (radius + height) Now, here's the thing. If you're not going to use this formula every day, there's absolutely no point in memorizing it. I certainly haven't! If I want to compute the surface area of a cylinder, I'll break it into two circles and a side, compute those areas, and add them up. And I recommend that you do the same thing, rather than trying to learn the compact formulas. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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