# Length of the Diameter of a Circle

The length of the diameter of the circle ${x^2} + {y^2} = {r^2}$ is equal to $2r$.

Consider the equation of a circle is given by

Let $A\left( {{x_1},{y_1}} \right)$ be the point of the circle. By putting this point in a circle, then

Since the origin $O\left( {0,0} \right)$ is the center of the given circle (i), the line through $O$ and $A$ across the circle is its diameter. The equation of the line through $O$ and $A$ using two points from the line is

Putting the value of $y$ from equation (iii) in equation (i), we have

Putting $x = - {x_1}$ in the above equation (iii), we have $y = \frac{{{y_1}\left( { - {x_1}} \right)}}{{{x_1}}} = - {y_1}$. This shows that the other end of the diameter is $B\left( { - {x_1},{y_1}} \right)$, as shown in the given diagram. Now the length of the diameter is

This is the required result.