# Thread: Cool Geometry Question

1. ## Cool Geometry Question

Ok this looks easy, but it isn't. To solve it, you need to obtain three independent equations containing x,y and z that can be solved to determine the areas of the regions.

A square has sides 6 cm long. Four quarter circles are inscribed in the square. Determine the areas of the three different kinds of regions that are formed.

Here's the picture:

So you need to make 3 equations with three unknown then solve. I happen to know two of them. They are:

4x + 4y + z = 36

This is because there are 4 "x" regions, 4 "y" regions, and 1 "z" region making up the square with area 6^2, aka 36.

2x+3y+z = 9pi

this is because there are 2 "x" regions, 3 "y" regions, and 1 "z" region making up a quarter circle with area (r^2 pi)/4 = (36pi)/4 = 9pi

What is the third equation? Its more difficult to find than the first two, I believe. Make sure your answer isn't just a combination of the first two, like 2x + y = 36 - 9pi

Thank you for all the help you can offer me.

2. Originally Posted by chris.j.stanford
Ok this looks easy, but it isn't. To solve it, you need to obtain three independent equations containing x,y and z that can be solved to determine the areas of the regions.

A square has sides 6 cm long. Four quarter circles are inscribed in the square. Determine the areas of the three different kinds of regions that are formed.

Here's the picture:

So you need to make 3 equations with three unknown then solve. I happen to know two of them. They are:

4x + 4y + z = 36

This is because there are 4 "x" regions, 4 "y" regions, and 1 "z" region making up the square with area 6^2, aka 36.

2x+3y+z = 9pi

this is because there are 2 "x" regions, 3 "y" regions, and 1 "z" region making up a quarter circle with area (r^2 pi)/4 = (36pi)/4 = 9pi

What is the third equation? Its more difficult to find than the first two, I believe. Make sure your answer isn't just a combination of the first two, like 2x + y = 36 - 9pi

Thank you for all the help you can offer me.
There is some kind of problem with the diagram, I can't see it.

RonL

3. ## Link fix

New Page 1

4. Area of square:

36 = z+4x+4y

Area of a quarter circle:

9 pi = z+2x+3y

Area of central lense shape:

2[18(pi/2-1)] = z+2y

You have three simultaneous equations in three unknows which you should be able to solve.

Except checking these are not linearly independent! So we still need another equation.

RonL

5. ## Hmmm

Yes, that problem is run into alot, the third equation is always sneakily just a combination of the other two. Thanks for trying, anyways... My teacher says it has something to do with symmetry.