Answer :

Answer:

To solve this problem, we'll start by finding the radius of the unit circle and then determine the scale factor needed to achieve the area of the new circle.

1. **Find the radius of the unit circle:**

The unit circle has a radius of 1.

2. **Area of the unit circle:**

The area of a circle is given by \( \pi r^2 \).

For the unit circle with \( r = 1 \):

\[

\text{Area} = \pi \cdot 1^2 = \pi

\]

3. **Find the radius of the new circle:**

We know the area of the new circle is \( 729\pi \) cm².

Using the area formula \( \pi R^2 = 729\pi \), we can solve for the new radius \( R \):

\[

R^2 = 729

\]

\[

R = \sqrt{729} = 27

\]

4. **Determine the scale factor:**

The scale factor \( k \) is the ratio of the new radius to the original radius.

\[

k = \frac{R}{r} = \frac{27}{1} = 27

\]

Therefore, the scale factor required to dilate the unit circle to a circle with an area of \( 729\pi \) cm² is 27.