How do you find the perimeter of a rectangle/square when you have the area of circles inside of it?

Question

I am stuck on a problem. There is a rectangle and four circles enclosed inside of it.  Thee problem gives me the area of the circles, but I do not know how to solve this problem based on this information.

Answer 1

Assuming the smallest rectangle, there are two options. Arrange the circles in a line, or in a 2 by 2 square arrangement.

 

Here they are:

 

OOOO

 

or

 

OO

OO

 

In the former you can see that the rectangle's width is four diameters, or 8r, and the height 2r for an area of 16r² and perimeter 20r. In the latter case the rectangle is a square of side 4r for the same area of 16r² and perimeter 16r.

 

The area of a circle is πr² so A_rectangle / A_circle = 16r² / πr² = 16/π

 

Hence we have A_rectangle = (16/π) A_circle

 

The perimeter is 20 / πr for the rectangle and 16 / πr for the square. shows shows it is not uniquely determined.

 

Now you can stick in the given area of the circle.

Answer 2

Hey Emma -- circles have about 75-80% of the area of squares [d*d] ... Acirc ~ 4/5 d*d

 

... area of your box is about 5/4 * 4 Acirc's ... or about 5 of the circles ... Regards :)

Answer 3

Emma,

 

I am trying to imagine this drawing they gave you.  If the circles are all in a line and touch one another, then the width of the rectangle is the diameter of the circle.  (I am assuming all the circles are of the same size .)  The length of the rectangle should be 4 times the diameter of the circle(s).

 

Since they give you are area of the circles I'm assuming you have to figure out what the diameter is.  You can do this easily by remembering

 

A = (pi)r²

 

r² = A/pi

 

r = √(A/pi)

 

and d (diameter) = 2r.

 

I hope this helps.  I wish this website would let us share diagrams.

Answer 4

You have the areas of the circles and you know that the area of one circle is Πr².  We can set the area of the circle to Πr² and solve for r and multiply r x 2 to find the diameter of the circle.  By finding the diameter of the circles you can figure out the base and height of the rectangle.  Since the area of a rectangle is base x height we can figure out the area of the rectangle given the area of the circles inscribed within the rectangle.