In order to solve this problem, what you need to know is how to find the circumference of a circle. The formula for circumference of a circle is
circumference = diameter*pi
Given that equation, you can see that the ratio of the circumferences will be the same as the ratio of the diameters. This is true because you are taking the two diameters and multiplying them by the identical number. This means the proportions will remain the same.
For example, if the diameters really were 1 and 3, the circumferences would be 3.14 (1*3.14 because pi = 3.14) and 9.42 (3*3.14). The ratio of these numbers is 1:3.
Suppose that the radius of circles are: D1 and D2.
We'll express the ratio of the 2 diameters as:
We know that the diameter of a circle is twice the radius of that circle, so:
The formula expressing the length of the circle is 2*pi*R.
The length of the first circle is: 2*pi*R1.
The length of the second circle is: 2*pi*R2.
The ratio of the circumferences of the 2 circles:
Simplifying constant pi and 2 , we'll get:
But R1/R2=1/3, as we've demonstrated before, so:
The cirumference of a circle is always pi times the diametrer.
Therefore, Circumference , C = Pi d, where d is the diameter of the circle.
Therefore, if C1 and C2 are the diameter of the two circles, with diameters d1 and d2, then
C1 = pi*d1.................(1) and
From (1) and (2) by dividing, C1/C2 = d1/d2 . But d1/d2 = 1/3 by the data.
Therefore, C1/C2 = 1/3 or C1: C2 = 1:3. So we arrived with proof that C1:C2, the ratio of the circumferences of the circle is 1:3