Constant property of a circle

The ratio of a circle’s circumference and its diameter is always a constant.

Introduction

To start studying the circles in geometry, everyone must learn an important property firstly. The ratio of the circumference of a circle to its diameter is a constant. It can be observed by comparing the ratios of two different circles. So, let’s construct two circles with different radii geometrically.

From a point, turn the thread around circle to reach the same point. After that, cut the thread and compare the length of the thread by a ruler. Remember, the length of the thread is the circumference of that circle.

You will observe that the length of the thread is equal to the $18.85 \,cm$ approximately. We know that the radius of the circle is $3 \,cm$. So, the diameter of the circle is equal to $6 \,cm$. Now, find their ratio mathematically.

$Ratio$ $\,=\,$ $\dfrac{18.85}{6}$

$\implies$ $Ratio$ $\,=\,$ $\dfrac{\cancel{18.85}}{\cancel{6}}$

$\implies$ $Ratio$ $\,=\,$ $3.1416666666\cdots$

$\,\,\,\therefore\,\,\,\,\,\,$ $Ratio$ $\,\approx\,$ $3.1417$

Repeat the same procedure for drawing another circle geometrically and calculate the ratio.

In this case, the radius of the circle is $4\,cm$. So, its diameter is equal to $8\,cm$. You will observe that the circumference of the circle is $25.15\,cm$ approximately. Now, calculate the ratio of the circle’s circumference to its diameter.

$Ratio$ $\,=\,$ $\dfrac{25.15}{8}$

$\implies$ $Ratio$ $\,=\,$ $\dfrac{\cancel{25.15}}{\cancel{8}}$

$\implies$ $Ratio$ $\,=\,$ $3.14375$

$\,\,\,\therefore\,\,\,\,\,\,$ $Ratio$ $\,\approx\,$ $3.1437$

Property

Compare the ratios for understanding the constant relation between the circumference of a circle to its diameter.

In the case of $3\,cm$ radius circle, the ratio of circle’s circumference and its diameter is $3.1417$
In the case of $4\,cm$ radius circle, the ratio of circle’s circumference and its diameter is $3.1437$

The two ratios are approximately equal. Even, you construct another circle with a different radius, you get the same value because the ratio of the circumference of a circle to its diameter is always constant. It is called the constant property of a circle.