Question

Derive the formula for the circumference of a circle.

Answer 1

Hint: Circumference of a circle is the measurement of the length of the boundary of the circle. Use the definition of \[\pi \], that is the ratio of the circumference to the diameter of any circle, to derive the formula of circumference of a circle.

Complete step by step solution:
The circumference of a circle is basically its perimeter or the length of the boundary of the circle. For easier understanding we can say that if a circle is cut open and made a straight line, then the length of this line is equal to its circumference.
To derive the formula for the circumference, we need to know the definition of \[\pi \] first.
By definition \[\pi \] is a mathematical quantity which is the ratio of the circumference of any circle to its diameter.
\[\therefore \pi = \dfrac{{circumference}}{{diameter}}\]
Let \[d\] be the diameter, \[r\] the radius and \[C\] the circumference of a circle,,
Now the diameter of any circle is twice its radius, \[\therefore d = 2r\],
\[\therefore \pi = \dfrac{C}{d}\]
\[\therefore \pi = \dfrac{C}{{2r}}\]
\[\therefore 2r\pi = C\]
Rearranging the above equation we get,
\[C = 2\pi r\].
This is the formula for the circumference of a circle.

Note: Remember this formula, as it is applicable for numerous problems related to circles. Try to observe from the formula that the circumference is directly proportional to the radius of the circle, that means if the radius increases the circumference also increases accordingly. Also be careful that the circumference means the perimeter of the circle and not its area.