Circle Area and Square Units

Date: 11/12/97 at 15:11:41
From: Sandy Miller
Subject: Circle area

Dear Dr. Math,

I can't figure out these two things and would be very grateful 
if you could help.

1) If the area if a circle with radius 1m is equal to pi metres 
   squared, then how come the same circle measured in centimetres 
   100cm has an area of 10,000 pi centimetres squared? Are these two 
   areas the same?

2) Does a square with sides of 10m have an area of 10m squared or 
   100 square metres?  Or are these the same?

Thank you very much.
Sandy Miller

Date: 01/14/98 at 13:04:11
From: Doctor Sonya
Subject: Re: Circle area

I think what is confusing you is the difference between calculating an 
area by squaring and getting an answer in units called square metres 
or square centimetres.  

When we measure a line, we can take a ruler and place it along the 
line.  If the line is curved we can lay a string along it and then 
lift it off - then pull it straight and use the ruler.

When we measure the area of a planar shape, we can't just use a ruler.  
Instead, imagine a clear plastic sheet with a grid of squares printed 
on it. We can count the number of squares that it takes to cover the 
shape and say the area is "so many square units."

If the shape is irregular, so the grid does not easily fit over it, we 
could cover the surface with sand or water to a uniform depth, say 1 
inch deep all over, then pour the sand or water into a rectangular 
tray to the same depth, and put our grid over the tray to get its 
area.  Tbis is kind of like laying a string along a curve and then 
measuring its length after you have straightened it out.

Some shapes like squares and circles occur so often that we have 
figured out formulas for their areas. These formulas involve linear 
measurements and squaring to give us square units. Square units are 
not at all like linear units of measure, but you can use them in the 
calculations.  

Now for your questions. In the first one, you asked, "If the area if a 
circle with radius 1m is equal to pi metres squared, then how come the 
same circle measured in centimetres 100cm has an area of 10,000 pi 
centimetres squared? Are these two areas the same?"

Area should be the same no matter which units we use to measure it.  
Say we use a ruler to measure the area of a square. No matter whether 
our ruler says centimeters or meters, the square doesn't change; only 
the number we use to describe it changes - but the two areas must be 
equal:

    pi square meters = 10,000 pi square centimeters

You know that 1 m = 100 cm. The formula for the area of a circle is:

   area = pi * r^2 

(r^2 is the length of the radius squared).

In meters, r = 1 m, and

   area = pi * r^2 
        = pi (1 m)^2 
        = pi * 1 m^2

(m^2 is another way to write square meters).   
        = pi m^2

In centimeters, r = 100 cm. Try to repeat the calculation I did above 
using this new value for r and see what you get.  

Now, because in both cases you are measuring the area of the same 
circle, are these two areas the same or different?

Now, for your second question. To answer this one, remember that 
square meters and meters squared are two ways to say the same thing.  
If you have a square that measures 5 m on each side, its area is:

    (5 m) * (5 m) = 25 m^2

which you can say as, "25 square meters" or "25 meters squared."


-Doctors Celko and Doctor Sonya,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/