Connection Between Circumference and Surface AreaDate: 05/08/98 at 16:26:53 From: MARGOT DIAZ LEARNED Subject: Derivation of the formula for surface area of a sphere I am helping a 7th grader with his math and we got to this section in his book about the formula for the surface area of a sphere. It said you could not intuitively figure out the connection between the area of a circle and the surface area of a sphere and that you could look up the derivation of the formula in other texts. Anyway, we got interested in it, but I don't have any other geometry texts lying about the house, so I came here and found that someone had asked a similar question for which they received an answer from Dr. Anthony, but the explanation is beyond my son, and I was wondering if you could explain it for the more mathematically illiterate, such as a 7th grader and a mom type? Date: 05/19/98 at 11:21:12 From: Doctor Jeremiah Subject: Re: Derivation of the formula for surface area of a sphere Hi Margot: You wanted to know about the connection between the area of a circle and the surface area of a sphere. Actually those two equations are not related. Instead the circumference of a circle has a connection to the surface area of a sphere and the area of a circle has a connection to the volume of a sphere. Circumference is the space taken up by the line around the outside of the circle and surface area is the space taken up by the outside of a sphere. That is why they are related to each other. The area of a circle is the space inside of the circle and volume of a sphere is the space taken up by the inside of a sphere. The circumference of a circle is: 2*Pi*R The surface area of a sphere is: 4*Pi*R^2 The area of a circle is: Pi*R^2 The volume of a sphere is: 4/3*Pi*R^3 To understand the relationship between the equations you need to notice that all of them involve the radius R. The way you can tell how many R's to include (or what power to raise R to) is by how many dimensions the object you are measuring has. When measuring the circumference of a circle, you are measuring the distance along a curved line, which is a one-dimensional curved line. If you look at the formula (2*Pi*R) there is only one factor of R in it, which corresponds with the one dimension. When measuring the area of a circle, you are measuring something in two dimensions, which corresponds with the two R's in the formula: Pi*R^2. It gets a little bit more complicated when we talk about surface area. While a sphere is definitely in 3 dimensions, its surface area is really only two-dimensional. Imagine measuring the surface area by wrapping the sphere in paper, and then cutting the paper off and flattening it out to measure how big it is. The surface area is two-dimensional, just like this paper. Now look at the formula for the surface area: 4*Pi*R^2 - we're still in two dimensions. So you can see that the difference is between related equations like circumference and surface area is that when you increase the number of dimensions from one to two or from you need to increase the exponent and find a new value for the coefficient at the front. To understand why the number in front changes, think about when a square changes to a cube and the perimeter of the square changes to surface area of a cube: Perimeter of a square = 4L (space around the outside of a square is 4 times the size of the side which is a line) Surface area of a cube = 6*L^2 (space around the outside of a cube is 6 times the size of the side which is a square) See how the number in front changes? This is something that always happens. To understand what the value should be in any particular situation requires calculus. Does that help? If you need more help please write again. -Doctors Jeremiah and Sonya, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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