Two circles have circumferences of π and 3π. What is the ratio

of the area of the circles? the diameters? the radii?

To find the ratio of the area of the circles, we need to determine the ratio of their radii squared.

The formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius. Given that one circle has a circumference of π and the other has a circumference of 3π, we can set up the following equations:

2πr1 = π
2πr2 = 3π

From the first equation, we can solve for r1:
r1 = π / (2π) = 1/2

From the second equation, we can solve for r2:
r2 = 3π / (2π) = 3/2

Now that we have the radii of the two circles, we can find the ratio of their areas. The formula for the area of a circle is A = πr^2. Therefore:

A1 = π(1/2)^2 = π/4
A2 = π(3/2)^2 = 9π/4

To find the ratio of the areas, we divide A2 by A1:
A2/A1 = (9π/4) / (π/4) = 9/1 = 9

Therefore, the ratio of the areas of the circles is 9:1.

To find the ratio of the diameters, we simply multiply the ratio of the radii by 2:
d1/d2 = (2r1) / (2r2) = r1 / r2 = (1/2) / (3/2) = 1/3

Therefore, the ratio of the diameters of the circles is 1:3.

Finally, we can find the ratio of the radii directly:
r1/r2 = (1/2) / (3/2) = 1/3

Therefore, the ratio of the radii of the circles is 1:3.