What is the ratio of the circumferences of two circles that have diameters in the ratio 1:5 ?
The formula for circumference of a circle is
C = pi.D
Given that pi is a constant, it follows that should the diameter is scaled, the circumference would follow proportionately.
That is to say, if the diameter of the second circle is five times of the first, so will its circumference be 5 times that of the first circle.
Let d and c be the diameter and circumference of the first circle;
and D and C be the diameter and circumference of the second circle.
c = pi * d
C = pi * D
But D = 5d
= pi * 5d
= 5 * pi * d
= 5 c
ie. Circumference of the big circle is 5 times the circumference of the small circle.
Since circumference is one-dimensional, the scaling will be in direct proportion.
However, this will not follow when it comes to area. For area, the scaling will be proportional to the square of the radius, since area is given by pi * r^2. Area of larger circle will then be proportional to the square of the radius.
We have to find the ratio of the circumferences of two circles which have diameters in the ratio 1:5.
Now we know that the radius is equal to half of the diameter.
If the diameters are in the ratio 1:5 the radius of the circles are in the ratio 1:5.
The circumference of a circle is given by 2*pi*r.
Let the radius of the circles be R1 and R2, we know R1: R2 = 1:5
Multiply R1 and R2 by 2*pi, we get 2*pi*R1 : 2*pi*R2 = 1:5.
Therefore the circumference of the two circles is in the ratio 1:5.
Suppose that the radius of circles are: D1 and D2.
We'll express the ratio of the 2 diameters as:
The diameter of a circle is twice the radius of the circle, so:
The length of the circle = the circumference of the circle = 2*pi*R.
The length of the first circle = 2*pi*R1
The length of the second circle = 2*pi*R2
The ratio of the circumferences of the 2 circles:
Simplifying constant pi and 2 , we'll get:
We can solve this problem in many different ways. The three answers posted below have used some of such methods to arrive at the correct answer. In addition to such mathematical methods, we can also use simple logic to come to the right answer without using any calculations at all.
Circumference of a circle is in direct proportion to its diameter. Therefore the ration of circumference of two circles will be same as the ratio of their diameters. Using this this logic we can directly say that:
Ratio of circumferences of two circles that have diameters in the ratio 1:5, is also 1:5.