What is the ratio of the circumferences of two circles that have diameters in the ratio 1:5 ?

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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...

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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.

To illustrate,

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.

Then,

c = pi * d

and

C = pi * D

But D = 5d

Hence,

C

= pi * 5d

= 5 * pi * d

= 5 c

ie. Circumference of the big circle is 5 times the circumference of the small circle.

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Note:

Since circumference is one-dimensional, the scaling will be in direct proportion.

WARNING:

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.

 

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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.

Approved by eNotes Editorial Team