Average area of image of rotated square

Suppose we have a unit square that is rotating in the x (or y) axis. At a point in time, it stops rotating. The image of it viewed from directly on top of it should be a rectangle. What is the average area of this rectangle?

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

If it rotates an angle of $\theta$ from the $xy$ -plane, then the area (viewed from above) will be $|\cos \theta|$ .

Therefore, the average area is $\dfrac{1}{2\pi}\displaystyle\int_{0}^{2\pi} |\cos \theta|\,d\theta = \dfrac{2}{\pi}$ .

Jimmy Kariznov - 7 years, 10 months ago

Can you rephrase it? I don't get how it's rotating...

Luca Bernardelli - 7 years, 10 months ago

Hmm... I'm not really sure how to say it. Imagine a square. You stick a pole through two opposite midpoints of the square, and rotate it with that pole as the axis. That kind of rotating.

Daniel Liu - 7 years, 10 months ago

How do you mean, it should be a rectangle? If you have a square, then you can rotate it all you want, but it stays a square, right?

Tim Vermeulen - 7 years, 10 months ago

But it is rotating in respect to the x axis, not the z axis.

Daniel Liu - 7 years, 10 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$