Question

How do you prove that the circumference of a circle is 2πr?

Answer 1

Hint:
It is a well-known fact that the ratio of the circumference of a circle to its diameter is fixed and its value is 3.14159... which is symbolically represented as π. Therefore, circumference = π × diameter = 2π × radius. We can prove that the ratio is the same constant for any circle, but we cannot prove why its value is what it is. It can only be "verified" by experiment that the value is equal to π = 3.14159...

Complete Step by step Solution:
Proving that the ratio of the circumference to the diameter is fixed for any circle:
This is quite easy to prove because all circles are similar.
Consider two small arcs AB and CD of two circles of radii x and y units, both of which make an angle of measure θ at the centers P and Q respectively.

For smaller values of θ, the arc length $ \overset\frown{AB} $ is almost equal to the length of the line segment $ \overline{AB} $. Similarly, $ \overset\frown{CD} $ ≈ $ \overline{CD} $ . Since the angle θ is the same in both the triangles APB and CQD, and the ratio AP : PB = CQ : QD = 1 : 1, using the SAS rule of triangle similarity, ΔAPB is similar to ΔCQD. Therefore, It follows that AB : AP = CD : CQ.
In other words, for all values of the angle θ, the ratios $ \overset\frown{AB} $ : AP and $ \overset\frown{CD} $ : CQ are equal. It follows that the ratio circumference : radius, which is equivalent to saying the ratio circumference : diameter is constant for any circle.
If this constant is equal to some number π, then circumference = π × diameter, or circumference = 2π × radius. Hence proved.

Note:
A circle is a geometric figure in which all the points are at a fixed distance 'r' from a fixed point 'O'. The distance 'r' is called the radius and the point 'O' is called the center of the circle.
The area of a circle with radius r units, is given by the formula: Area = $ \pi {{r}^{2}} $ square units.