Area of a Circle with Radius less than 1

Date: 02/18/2002 at 19:11:11
From: Kelly
Subject: Finding the Area od a circle with a radius less than 1

If the radius of a circle is less than 1, for example half an inch, it 
just gets smaller and you get a smaller area.
 
I was doing a report on finding household objects and their 
circumference, area, etc., so I used inches on one small object, 
making it have radius less than 1. Then I tried with centimeters and I 
found that when I compared centimeters to inches by seeing how many 
centimeters fit into an inch, I found the length with the centimeters 
was longer because it was not less than 1.

Please help!  Thank you.
-Kelly

Date: 02/18/2002 at 19:29:15
From: Doctor Ian
Subject: Re: Finding the Area od a circle with a radius less than 1

Hi Kelly,

It doesn't really make sense to talk about a 'radius less than 1'.  
You need to specify some kind of unit of length: centimeters, inches, 
miles, meters, light years, and so on.  

If you pick the right units, you can make the radius of just about 
anything either less than or greater than 1.  

Consider a car tire. When measured in inches, the radius is clearly 
greater than 1. But when measured in miles, the radius is clearly less 
than 1. How can this be possible? 

The confusion clears up when you attach the units. It isn't surprising 
at all to say that something is 'larger than 1 inch but smaller than 1 
mile', or 'larger than 1 foot but smaller than 1 yard'.  Is it? 

If the point is to find areas of objects with radius less than 1, with 
no particular unit specified, then you can just choose a different 
unit for each object. When measuring a dinner plate, use yards. When 
measuring a quarter, use feet. When measuring a shirt button, use 
inches. And so on. Or just pick a really big unit, like yards, and 
measure everything in yards. You'll end up with some pretty silly 
numbers, i.e., 

  a coat button is 1/2 inch in radius.  So its radius in yards is

                  1 ft    1 yard
   r = (1/2) in * ----- * ------ = 1/72 yards
                  12 in   3 ft

  so the area is 

   a = pi * r^2 

     = (22/7) * (1/72)^2

     = 22/36288 square yards

Does this help?  Write back if you'd like to talk more about this, or 
anything else. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 02/19/2002 at 19:20:04
From: Kelly
Subject: Finding the Area of a circle with a radius less than 1

I don't know if that really answered my question. What I was trying to 
get at is that if a number is less than 1, it will be smaller than it 
is supposed to be. I think that 2.5 centimeters approximately equals 
1 inch, so here is my example of what I mean: if you use a unit that 
is the same measurement but smaller, if you use something over 1 it 
will be larger even though it was the same at first - if that 
doesn't make sense look at my example.

There is a circle. The Diamater is 1 inch, or 2.5 centimeters, and you 
want to find the area, so the radius is half an inch (.5 inch) or 1.25 
cm. This is the work i did for each unit:

Inches:
Radius = .5 inch (.5 times .5 = .25) (.25 times 3.14 = .785) .785 
inches transferred into centimeters should be around 1.9 (1.9625 
exactly I think) but it is not, so if you do the centimeter work

Centimeters:
Radius = 1.25 cm (1.25 times 1.25= 1.5625) (1.5625 times 3.14 = 
approximately 4.9) 4.9cm is much more than the original 1.9 you would 
find.

So I guess what I was wondering is why is this true? If a number is 
under 1 should you not square it? Is there some kind of formula 
explaining this? Is there something you can do to make me understand 
if there is some reason for this and if I am doing something wrong? 

Please reply! Thank you.
-Kelly

Date: 02/20/2002 at 12:33:05
From: Doctor Ian
Subject: Re: Finding the Area of a circle with a radius less than 1 

Hi Kelly,

I think perhaps the problem is that you're confusing linear units 
(inches, centimeters, feet) with squared units (square inches, square 
centimeters, square feet).

To use your example, if I have a circle whose radius is 1/2 inch, the 
area is 

  a = pi * (1/2)^2

    = (22/7)(1/2 in)(1/2 in)

    = 11/14 in^2

which is about 0.785 in^2.  

The exact conversion between inches and cm is 1 in = 2.54 cm, so when 
we use cm as the units, we find an area of 

  a = pi * (2.54/2)^2

    = (22/7)(2.54/2 cm)(2.54/2 cm)

    = (22 * 2.54 cm * 2.54 cm) / (7 * 2 * 2)

    = 5.07 cm^2

1.9 in^2 (which is another way to write '1.9 square inches').  

So far, we're in agreement. And these two areas should be the same, 
right?  That is, since it's the same circle, it ought to be true that

  0.785 in^2 = 5.07 cm^2

and if I understand you correctly, you want to know (1) whether this 
is true, and if so, (2) why it's true. 

Well, let's look at the area of a square that is one inch on each 
side:

    1 in
  +------+
  |      | 1 in       area = 1 in * 1 in 
  |      |
  +------+                 = (1 * 1) in^2

                           = 1 in^2

If we measure the sides in centimeters instead of inches, we have

    2.54 cm
  +------+
  |      | 2.54 cm       area = 2.54 cm * 2.54 cm
  |      |
  +------+                    = (2.54 * 2.54) cm^2

                              = 6.45 cm^2

That is, while it's true that 

  1 in = 2.54 cm

it's NOT true that

  1 in^2 = 2.54 cm^2       (Wrong!)

Rather, the conversions are

  1 in   = 2.54 cm

  1 in^2 = 6.45 cm^2

This might make more sense if we forget about the metric system for a 
moment.  Let's look at square that is 1 yard on a side:

        1 yd
  +----|----|----+
  |              |
  -              -
  |              | 1 yd
  -              -
  |              |
  +----|----|----+

The area is clearly 1 yd^2.  But if we divide it up into square feet, 

   1 ft 1 ft 1 ft
  +----+----+----+
  |    |    |    | 1 ft
  +----+----+----+
  |    |    |    | 1 ft
  +----+----+----+
  |    |    |    | 1 ft
  +----+----+----+

we just as clearly have an area of 9 ft^2. Does this make sense? (It's 
easier to see what's going on in this case because it deals with whole 
numbers instead of fractions or decimals.)  So the conversions are NOT

  1 yd   = 3 ft

  1 yd^2 = 3 ft^2    (Wrong!)

Rather, the conversions are

  1 yd   = 3 ft

  1 yd^2 = 9 ft^2 

Now let's get back to your original question. You want to know whether 
it's true that

  0.785 in^2 = 5.07 cm^2

Well, let's use the conversion that we found, 1 in^2 = 6.45 cm^2:

               6.45 cm^2
  0.785 in^2 * --------- = 5.07 cm^2
               1 in^2


               5.06 cm^2 = 5.07 cm^2

which isn't exact, but the difference can be accounted for by the fact 
that we rounded off our decimal approximations.  

This is a subtle concept, and a lot of people have trouble with it, so 
don't feel too bad about being confused!  You might want to take a 
look at 

   Area and Perimeter
   http://mathforum.org/dr.math/problems/jessica.5.1.01.html   

to get a feel for how length and area differ from each other. 

I hope this helps.  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/