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Radius of a Sphere


Date: 01/29/98 at 00:27:35
From: Nino John A. Guerrero
Subject: Geometry

The radius of the sphere in the exercise is 3 times the radius of the 
sphere in the examples. What is the ratio of the areas of the two 
spheres? What is the ratio of their volumes? What possible theorems 
are suggested by the results?


Date: 01/30/98 at 17:37:04
From: Doctor Bill
Subject: Re: Geometry

Nino,

If the sphere in the examples has a radius of r, then the sphere in 
the exercise has a radius of 3r, since it is 3 times larger.

The formula for the surface area of a sphere is 4*pi*r^2, which will 
be the area of the sphere in the examples. Since the radius of the 
sphere in the exercise is 3r, then its surface area is 4*pi*(3r)^2 = 
4*pi*9r^2. To find the ratio of the surface areas of the two spheres 
we divide one formula by the other: (4*pi*9r^2)/(4*pi*r^2). 

After you simplify by cancelling the common terms in the numerator and 
the denominator, you should get that the ratio is 9. That means if you 
increase the radius of a sphere 3 times, you increase the surface area 
9 times. Do you see a realationship between the 3 and the 9? 

What if you increased the radius by 4 times, that is, the new radius 
is 4r - how much do you think the surface area would increase? (As a 
hint, notice that in the formula the radius is SQUARED.) Check it out 
to see by plugging 4r into the formulas above where we had 3r before.

The formula for the volume of a sphere is (4/3)*pi*r^3, which is the 
volume of the sphere in the examples. Once again, the radius of the 
sphere in the exercise is 3 times larger, so its volume is (4/3)*pi*
(3r)^3. Now divide these two formulas as we did above and simplify. 
You should get that the larger sphere (with radius 3r) has a volume 
27 times larger than the smaller sphere. 

Once again, do you notice a relation between the 3 and the 27? What 
do you think the ratio of the volumes would be if you increased the 
radius by 4 times, that, is make the new radius 4r? (Another hint, 
notice that in the volume formula the radius is raised to the 3rd 
power.)

From the two results above, and the questions I asked, what two 
theorems do you think you can state now?

-Doctor Bill,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry
Middle School Geometry
Middle School Higher-Dimensional Geometry
Middle School Ratio and Proportion

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