The ratio of the radii of two circles is 2:3.
find the ratio of their circumferences and the ratio of their areas
Circumference is a linear measurement, so
the ratio is still 2:3
but...
areas of circles are proportional to the square of their radii.
so 2^2 : 3^2 = 4 : 9
Show me the process
To find the ratio of the circumferences, we need to know that the formula for the circumference of a circle is C = 2πr, where C is the circumference and r is the radius.
Given that the ratio of the radii is 2:3, let's assume the radii of the two circles are 2x and 3x, respectively.
The circumference of the first circle (C1) with radius 2x can be calculated as C1 = 2π(2x) = 4πx.
The circumference of the second circle (C2) with radius 3x can be calculated as C2 = 2π(3x) = 6πx.
Therefore, the ratio of the circumferences (C1:C2) is 4πx:6πx, which simplifies to 2:3.
Now let's find the ratio of their areas:
The formula for the area of a circle is A = πr^2, where A is the area and r is the radius.
Using the previously assumed radii of 2x and 3x for the two circles:
The area of the first circle (A1) with radius 2x can be calculated as A1 = π(2x)^2 = 4πx^2.
The area of the second circle (A2) with radius 3x can be calculated as A2 = π(3x)^2 = 9πx^2.
Therefore, the ratio of the areas (A1:A2) is 4πx^2:9πx^2, which simplifies to 4:9.
To summarize:
- The ratio of the circumferences of the two circles is 2:3.
- The ratio of the areas of the two circles is 4:9.