The radii of two circles are in the ratio of 3 to 1. Find the area of the smaller circle if the area of the larger circle is 27 sq. in.

To find the area of the smaller circle, we first need to determine the ratio of their radii. The given information states that the radii of the two circles are in the ratio of 3 to 1.

Let's assume that the radius of the larger circle is 3x (multiplying both parts of the ratio by a factor of 3) and the radius of the smaller circle is x.

The area of a circle can be calculated using the formula A = πr², where A is the area and r is the radius.

For the larger circle with radius 3x, we have:
A1 = π(3x)²
= π(9x²)

Given that A1 (the area of the larger circle) is 27 sq. in., we can equate it:
π(9x²) = 27

Dividing both sides of the equation by 9π:
x² = 3

Taking the square root of both sides:
x = √3

Now that we have found the value of x, we can calculate the area of the smaller circle with radius x:
A2 = π(x)²
= π(√3)²
= π(3)

Hence, the area of the smaller circle is 3π or approximately 9.42 sq. in.

a/27 = (1/3)^2