Question

derive the formula for the circumference of a circle with radius ( r ) by following these steps.

1. because all circles are similar, set up a proportion of the circumference to the diameter for the two circles shown in the diagram.

\frac{c_1}{d_1} = \frac{c_2}{d_2}

1. suppose ( d_1 = 1 ). by definition, what is the circumference of the smaller circle? ( c_1 = )
2. suppose the radius of the larger circle is ( r ). what is its diameter? ( d_2 = )

options: ( r/2 ), ( r ), ( 2r )

Explanation:

Step 1: Recall the definition of diameter

The diameter of a circle is twice the radius. For a circle with radius \( r \), the diameter \( d \) is given by the formula \( d = 2r \).

Step 2: Apply to the larger circle

The larger circle has a radius of \( r \). Using the diameter - radius relationship (\( d = 2r \)), the diameter \( d_2 \) of the larger circle is \( 2r \).

Answer:

\( 2r \)