25 Area and circumference of Circles Critical Thinking Solve each problem. 1. Suppose you know that the radius of one circle is three times the radius of another circle. Explain how you can determine the ratio of the areas of the circles and the ratio of the circumferences of the circles without knowing the actual measurements.

To find the ratio of the areas of the circles, remember that the area \( A \) of a circle is calculated using the formula \( A = \pi r^2 \). If one circle has a radius \( r \) and the other has a radius \( 3r \), the areas become \( \pi r^2 \) and \( \pi (3r)^2 = 9\pi r^2 \). Thus, the ratio of the areas is \( 1:9 \)! For the circumference \( C \), which is given by the formula \( C = 2\pi r \), the circumferences become \( 2\pi r \) for the first circle and \( 2\pi (3r) = 6\pi r \) for the second circle. Consequently, the ratio of the circumferences is \( 1:6 \)!