Waste less time on Facebook — follow Brilliant.
×

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?

Note by Daniel Liu
4 years, 6 months ago

No vote yet
1 vote

  Easy Math Editor

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. list

  1. numbered
  2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1

paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
    # 4 spaces, and now they show
    # up as a code block.

    print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears 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

Sort by:

Top Newest

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 - 4 years, 5 months ago

Log in to reply

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 - 4 years, 5 months ago

Log in to reply

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

Daniel Liu - 4 years, 5 months ago

Log in to reply

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

Luca Bernardelli - 4 years, 5 months ago

Log in to reply

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 - 4 years, 5 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...