Painting a Merry-Go-Round

Date: 06/06/2002 at 13:12:43
From: Eric V
Subject: Area of Circle from Inscribed Circle Tangent Line

I heard a math question on the radio, strangly enough.  I'm an 
engineering major, and am sort of embarassed I can't figure this out, 
but it's been driving me crazy.  The problem is as follows:

"A man has a merry-go-round that he needs to paint.  He has painted 
all of it but the floor where the horses are.  He removed all the 
horses, so it's just a big circle, with a column in the center where 
the machinery is.  He wants to know the area of the floor (not 
including the inscribed circle/column) so he knows if he has enough 
paint.  So, to recap, there is basically a circle (of undetermined 
size), with another circle (of undetermined size) inscribed in it.  
The only measurement you are given is a chord that is tangent to the 
inscribed circle, the measurement being 70 feet.  Find the area of 
the floor, not including the inscribed circle."

I can't figure out how to solve this, but I'm sure it has an answer.  
Any help you could give me would be very appreciated, thanks.

Date: 06/07/2002 at 12:46:32
From: Doctor Ian
Subject: Re: Area of Circle from Inscribed Circle Tangent Line

Hi Eric,

Call the radius of the larger and smaller circles R and r, 
respectively.  Then the area of the part to be painted is 

  pi * R^2 - pi * r^2 = pi(R^2 - r^2)

Here's a hint:  Draw a picture of the situation, and look for a 
right triangle whose sides are R, r, and 70/2. 

Can you take it from here?

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

Date: 06/07/2002 at 14:52:18
From: Eric V
Subject: Area of Circle from Inscribed Circle Tangent Line

I'm afraid that doesn't help.  I worked on this for a while and made 
all sorts of triangles, including the kind you talked about; the 
right traingle with sides R, r, and 70/2.  However, this doesn't 
give you an answer (you can't solve for two variables with one 
equation). 

When I worked on it to find the second equation, I kept 
getting a theta that I couldn't (A) solve for, or (B) cancel out.  
I'm trying to find the actual numerical values of the radii, not 
just how they relate to each other.  

--Eric

Date: 06/07/2002 at 15:14:58
From: Doctor Ian
Subject: Re: Area of Circle from Inscribed Circle Tangent Line

Hi Eric,

The key to solving this is to recognize that the problem doesn't 
ask you to figure out the areas of the circles.  And it's a good 
thing, because there are infinitely many pairs of circles that 
fit the description.  

The problem asks you to figure out the _difference_ of the areas. 

(You might want to take another crack at the problem before 
reading further.)

Draw the two circles, and the chord.  Label the ends of the chord 
A and B, and the point where the chord is tangent to the inner 
circle C.  Label the (shared) center of the circles D.  

Now look at the triangle DCB.  It's a right triangle.  The 
hypotenuse (DB) is R, the shorter leg (DC) is r, and the 
remaining leg (CB) is half the length of the chord, in this case, 
35 feet. 

The Pythagorean theorem tells us that

  R^2 = r^2 + 35^2

which means that 

   R^2 - r^2 = 35^2

But the area that needs to be painted is pi(R^2 - r^2), so 

  area to be painted = pi(R^2 - r^2) = pi * 35^2

There's no way to pin down the actual radii of the circles, but 
for _any_ two concentric circles where a chord of length L is 
tangent to the smaller circle, the difference in areas is 
pi*(L/2)^2.  

Isn't that wild? 

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

Date: 06/08/2002 at 12:45:00
From: Eric V
Subject: Thank you (Area of Circle from Inscribed Circle Tangent Line)

Wow.  Thanks a lot, that problem was driving me crazy.