**Finding the relationship between circumference and area :**

We can use what we know about circumference and area of circles to find a relationship between them.

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

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

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

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

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