To do this (you don't have to worry about the details of what I'm doing) I want to draw a target with two concentric circles. Here's the tough part that I need you to help me with. If the outside circle is 36 inches in diameter, how big should the inner circle be so that the area of the inner circle equals the area of the outer circle NOT covered by the inner circle? So if the inner circle is black and the rest of the outer circle is white, the white area equals the black area.

(I think I explained that okay - just read it carefully and draw yourself a picture.)

This problem turned out to be a little bit easier than I originally thought. 177 people got it right, and 64 got it wrong. The reasons behind the wrong answers ranged from math errors, to using the wrong formula for area, to just figuring it had to be half the radius of the other one. A couple of those types of errors could be caught if you checked your answer when you were done - find the areas and see if they have the right relationship. Never assume you did everything right. Still other people didn't provide an explanation. You will NOT get credit if you don't provide an explanation!

I thought this would be harder because the way I did it was to set up an equation where the area of the outside minus the area of the inside equals the area of the inside. This works just fine, but I thought it might leave some room for math errors and problems figuring out exactly what should go where. Debra Goldenberg of Marple Newtown High School provided a nice example of this, which I have included below. About 30 people used this method.

The other method (they aren't really different, but the first one looks more complicated) was to find the area of the outer circle and divide it by two. Then give the answer from there. Justin McVicker of Livermore High School did it this way. You can read his solution below, and note that he gave the radius, diameter, and circumference. I didn't say which dimension I wanted, I just asked, "How big is the inner circle?" That gave you the freedom to choose, though I guessed a lot of people might give radius, since that is the dimension you were given for the first circle.

Jen Erhart of Shaler Area High School solved it the "second" way as well, and provided a very thorough answer. She even checked it, and explained how to figure it out based on a starting circle of any size. Her solution is included.

Kevin Markham of Murphy High School solved it both ways and did a really nice job, so I included his as well.

And one more time, all together: Pie is a dessert; pi is a Greek letter. Please don't use the pi character on your keyboard - it won't look the same when it gets to me. Jordie Kvidera of Olde Middle School used two capital t's for the pi symbol - TT - and I think that worked pretty well.

In case you were wondering, let me tell you a little bit about how I would use this target with my shotgun. A shotgun shoots a bunch of little pellets, and the "load" (the pellets) spreads out the farther it gets from the gun. When you "pattern" a shotgun, you shoot it at paper to see if it spreads out evenly. So I will take this target that you helped me design and add a vertical and a horizontal line through the center. Now I have a target with 8 regions of equal area. I'll count how many pellets are in one of my shotgun shells and then shoot it at the center of the paper. Then I can count how many pellets hit in each region. It's kind of tedious, but it tells you a lot about your gun, and there are things that you can do to make it shoot a better pattern. Thanks for your help!

A list of all the people who got this problem right follows the highlighted solutions below, and most of the solutions are also available.

From:   Justin McVicker
        
Grade:  9
School: Livermore High School, Livermore, California

OK. The first thing you need to do is find the area of the large
circle. To do this, you use the formula pi times radius squared.
Since we know the diameter of the large circle is 36, the radius
is 18. Plugging 18 into the equation, we get about 1,017.36 as
the area. 

Now, since the small circle and the large circle uncovered must
be equal, that means that the area of each must be the area of
the entire large circle divided by 2, which is about 508.68.

Next, we sort of reverse the equation for the area of a circle
to get the radius of the small circle. We take the area (508.68)
and divide it by pi, giving us 162. Then we square root 162,
which gives us about 12.73. 

So, the radius of the small circle is 12.73, the diameter is
about 24.46, and the area is about 508.68

From:   Kevin Markham
        
Grade:  12
School: Murphy High School, Mobile, Alabama

Let r = radius of the inner circle. The general formula for the area of a circle 
is A = (pi)r^2, so the area of the inner circle will be (pi)r^2.

Since the diameter of the outer circle is 36, its radius will be 18. Thus, its 
area will be (pi)(18)^2, or 324(pi).

Since the inner circle and the ring around it have the same areas, the area of 
the ring is also (pi)r^2. Thus, the area of the outer circle is 2(pi)r^2, or the 
sum of the areas of the inner circle [(pi)r^2] and the ring [(pi)r^2].

Since the area of the outer circle can be expressed as either 324(pi) or 2(pi)r^
2, they can be set equal to each other, allowing you to solve for r as follows.
2(pi)r^2 = 324(pi)
    2r^2 = 324
     r^2 = 162
       r = sqrt(162)
       r = 9*sqrt(2)

This can be verified by checking that the sum of the areas of the inner circle 
and the ring is equal to the area of the outer circle.
A(inner) = (pi)r^2 = (pi)[9*sqrt(2)]^2 = 162(pi)
A(ring) = A(inner) = 162(pi)
A(ring)+A(inner) = 162(pi)+162(pi) = 324(pi)
A(outer) = 324(pi)

This problem could also be solved by stating that:
1. the area of the inner circle equals the area of the outer circle minus the 
area of the ring
    (pi)r^2 = 324(pi)-(pi)r^2
   2(pi)r^2 = 324(pi)
2. or that the area of the inner circle equals one-half the area of the outer 
circle.
    (pi)r^2 = 324(pi)/2
   2(pi)r^2 = 324(pi)

From:   Debra Goldenberg
        
Grade:  10
School: Marple Newtown High School, Newtown Square, Pennsylvania

I set up a drawing showing the diameter and radius of the circles.  Using the 
formula A=r2pi (sorry no key for it on my keyboard).  I set up an equation to 
show the relationship between the area of the inner circle and the area of the 
outer circle not covered by the inner circle.  pi182-x2pi=x2pi.  Then I solved 
for x (x is the radius for the inner circle). 
pi182-x2pi=x2pi
     +x2pi +x2pi
pi324=2x2pi 
Then I divided by 2pi
162=x2
x=9root2
The inner circle should be 18root2 inches in diameter.

From:   Jen Erhart
        
Grade:  
School: Shaler Area High School, Pittsburgh, Pennsylvania

Subject: pow

Hi.  This is Jen Erhart from Mr. Lishack's geometry class at Shaler.

The Problem of the Week deals with two concentric circles.  Concentric circles 
have the same center.  The problem asks if the outside circle is thirty-six 
inches in diameter, how large should the inner circle be so the area of the 
inner circle is equivalent to the area of the outer circle not covered by the 
inner circle.
	To begin this problem, I found the area of the outer circle.  The diameter 
of a circle is equal to the radius multiplied by two.  Therefore, the radius is 
18 inches.  The equation for finding the area of a circle is pi(r^2).  Thus, if 
18 inches is substituted in for r, the area of the circle is 1017.87602 square 
inches.  
	I then realized that is if this area is divided in half and used as the area 
of the smaller circle, the area of an inner circle would be equivalent to the 
area of the larger circle not covered by the smaller circle.  Thus, I set up the 
equation pi(r^2)=1017.87602/2.  Simplified this equation is pi(r^2)=508.9380099 
square inches.  If 508.9380099 square inches is divided by pi, the equation 
becomes r^2=162 square inches.  If the square root of both sides is taken, r=
12.72792206 inches.  This is the radius of the smaller, inner circle.  
	The area of the smaller circle can be found by using the equation for the 
area of a circle, A=pi(r^2).  If 12.72792206 inches is substituted for r, the 
area of the inner circle is 508.9380098 square inches.  To determine if the area 
of the outer circle not covered by the inner circle is equal to the area of the 
inner circle, the area of the inner circle can be subtracted form the area of 
the larger circle.  Therefore, 1017.87602 square inches-508.9380098 square 
inches should equal the area of the inner circle.  By doing the subtraction it 
is shown that it does, because the area of the outer circle not covered by the 
inner circle is 508.93801 square inches.
This method can be used for circles of any size.  All that must be done is to 
find the area of the larger circle using the equation A=pi(r^2).  If x is 
substituted in for r, the equation becomes A=pi(x^2).  This area is then divided 
by two, to give pi(x^2)/2.  This equation is then set up to the equation for the 
area of the inner circle, or pi(r^2).  Therefore, the new equation is pi(r^2)=
pi(x^2)/2.  The pi on both sides cancels out so that the equation is r^2=x^2/2.  
If the square root of both sides is taken, the equation becomes r=x/square root 
of two.  If the fraction on the right side of the equation is multiplied by the 
square root of two/square root of two, the equation is then x(square root of 
two)/2=r, where x is the radius of the larger circle, and r is the radius of the 
smaller circle.  This equation can be used for circles of any size.

The following students submitted correct solutions this week: