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### 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.
```

```
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

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.
```
Associated Topics:
High School Conic Sections/Circles

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