FINDING THE RELATIONSHIP BETWEEN CIRCUMFERENCE AND AREA

Start with a circle that has radius r.

We know that the formula to find circumference and area of a circle that has radius r. 

Circumference  :  C = 2πr

Area  :  A = πr²

Step 1 : 

Solve the equation C = 2πr for r.

Divide both sides by 2π.

C/2π  =  2πr/2π

C/2π  =  r

Step 2 : 

Plug r  =  C/2π in the formula for area of a circle. 

A = π(C/2π)²

Square the terms inside the parenthesis. 

A = πC²/4π²

Simplify 

A = C²/4π

Step 3 :

Solve the above equation for C². 

Multiply both sides by 4π.  

4πA = 4π(C²/4π)

Simplify

4πA =  C²  or  C² =  4πA

So, the circumference of the circle squared is equal to four times π times the area. 

Reflect

Does this formula work for a circle with a radius of 3 inches ? Show your work.

Answer : 

Yes

Since the radius 3 inches is not a multiple of 7, we can use π  =  3.14

Step 1 :

Find area of the circle. 

A  =  πr²

Plug r  =  3

A  =  π(3)²

A  =  9π

Plug  π  ≈  3.14

A  ≈  9(3.14)

A  ≈  28.26

Step 2 : 

Find circumference of the circle. 

C  =  2πr

Plug r  =  3

C =  2π(3)

A  =  6π

Plug  π  ≈  3.14

C  ≈  6(3.14)

C  ≈  18.84

Step 3 : 

Square the circumference of the circle. 

C²  =  (18.84)²

C²  =  354.9456 -----(1)

Multiply the area by 4π. 

4πA    ≈  4 x 3.14 x28.26

4πA    ≈  354.9456 -----(2)

From (1) and (2), we get

C²  =  4πA

So, the circumference of the circle squared is equal to four times π times the area. 

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