I would like to find the average chord length of a circle.(adsbygoogle = window.adsbygoogle || []).push({});

And I have 2 methods, which gave different answers...

[The chord is defined as the line joining 2 points on the circumference of the circle.]

The general formula for a chord length is ##d=2R\sin(\delta/2)=2\sqrt{R^2-u^2}##

Method 1: Integrate over angles

Avg length = $$\frac{\int_0^{2\pi} 2R\sin(\delta/2)\,d\delta}{2\pi}$$

$$=\frac{4R}{\pi}$$

Method 2: Integrate over diameter (-R to R)

Avg length = $$\frac{\int_{-R}^{R} 2\sqrt{R^2-u^2}\,du}{2R}$$

After simplification

$$\int_{-R}^{R} \sqrt{1-\left(\frac{u}{R}\right)^2}\,du$$

Then, using the fact

$$\int_{-Z}^{Z} \sqrt{1-\left(\frac{u}{Z}\right)^2}\,du = \frac{Z\pi}{2}$$

I get

$$=\frac{R\pi}{2}$$

So the answers from my 2 methods dont add up. Any things i might have overlooked?

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# I Average chord length of a circle

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