# Determining the Diameter of a Circle

The radius of a circle can in one way be determined if we know its Diameter $D$. It can be determined by the equation:

$r = \frac D2$ ..... (I)

We can also determine the radius of a circle by Euclidean distance formula which is as follows if the centre of a circle is at origin:

$\sqrt{x^2 + y^2} = r$ ..... (II)

If we compare the equations (I) and (II), then we get:

$\frac D2 = \sqrt{x^2 + y^2}$

OR

$D = 2(\sqrt{x^2 + y^2})$

Is it allowed to establish an equation to determine the Diameter of a circle like the above?

• I don't understand the question what is 'it' and what are we trying to establish? – Ethan Apr 21 '13 at 7:18
• The question is kind of confusing. Given a point $(x,\ y)$, the equation $\sqrt{x^2 + y^2} = r$ gives the radius of the circle centered on the origin, passing through $(x,\ y)$. Then $2r=d$ is indeed the diameter of the circle. Are you simply asking if this is a valid train of thought? – EuYu Apr 21 '13 at 7:20
• @EuYu yes! I'm asking that. – Samama Fahim Apr 21 '13 at 7:25
• @SamamaFahim Then yes, it is valid. – EuYu Apr 21 '13 at 7:28

Yes is valid. I don't know where is your confusion but notice that $x^2+y^2\geq 0$. Therefore $\sqrt{x^2+y^2}$ is always well defined in the sense that you get a real number as a result, and of course is legal two divide by two. So the expression is mathematically valid and you are able to calculate it for any point $(x,y)$ of the space. The reasoning of deduction of the formula is also flawless.