+0  
 
0
37
2
avatar

A square and a circle have the same perimeter.  Find the ratio of the area of the square to the area of the circle.

 May 16, 2020
 #1
avatar+20798 
+2

Since we are finding a ratio, we can choose whatever we want to be the perimeter of the square.

I'm going to choose 40; this makes each side of the square equal to 10 and the area equal to 100.

 

Since the circle has the same perimeter (circumference), I'm going to use the formula for the circumference of the circle to find the radius of the circle.

 

C  =  2·pi·r     --->     40  =  2·pi·r     --->     40 / (2·pi)  =  r     --->     r  =  6.366  (approximately)

 

The area of the circle is:  pi·r2  =  pi·6.3662  =  127.3  (approximately)

 

The ratio of the area of the square to the area of the circle is:  100 / 127.3  =  0.786    (approximately)

 May 16, 2020
 #2
avatar+111360 
+1

Thx, geno......here's another way

 

Call  the perimeter of the  circle  2pi * r

 

Then the side of the square  must  be      [ 2pi * r]  / 4    =    [ pi * r ]  2

 

Then the  area  of the square  =  pi^2 r^2 / 4

 

And the area of the circle is pi * r^2

 

So....the ratio of the area of the square to the area of the circle is :

 

pi^2 * r^2  / 4                pi

___________   =      _____   ≈  .785

     pi * r^2                     4

 

 

cool cool cool

 May 16, 2020

23 Online Users

avatar