Question

given two circles (all circles are similar) , with circumferences of 30cm and 12cm each, find the ratio of their areas. state answer as fraction.

Answer 1

The circumference of a circle is given by the following formula

`C=2\pi r`

where r represents the radius.

The ratio between two circumferences is equal to the ratio of the radius.

`(C_1)/(C_2)=(2\pi r_1)/(2\pi r_2)=(r_1)/(r_2)`

The area of a circle is given by the following formula

`A=\pi r^2`

Then, the ratio between two circle areas is equal to the square of the ratio of the radius, which is the square of the ratio between the circumferences.

`(A_1)/(A_2)=(\pi r_1^2)/(\pi r_2^2)=((r_1)/(r_2))^2=((C_1)/(C_2))^2`

Then, applying this relation in our problem, the ratio between the areas is:

`(A_1)/(A_2)=((30)/(12))^2=(25)/(4)`

The ratio between the areas is 25/4.