• 0

Triangles inside circles

Question

Posted · Report post

A few puzzles posted in this forum have related to random triangles inside a circle.

By evaluating nasty integrals, or by my preferred method, simulation, it can be shown, perhaps surprisingly, that triangles constructed from sets of three uniformly chosen points within a circle cover only about 7.388% of the circle's area on average. After looking at on the subject, I simulated 1 million triangles to determine the median area. It turns out to be about 5.335% of the circle's area. Read: a random triangle has a 50% chance of being smaller.

If the distribution of random-triangle areas has a mean of about 7.4% and a median of about 5.3%, what value might you expect for the mode?

0

Share this post


Link to post
Share on other sites

5 answers to this question

  • 0

Posted · Report post

For mode:

Divide the area range from zero to the maximum possible area (inscribed equilateral triangle) into small intervals.

Which interval would contain the most area values for a large number of randomly drawn triangles?

0

Share this post


Link to post
Share on other sites
  • 0

Posted · Report post

around 5.1% ??

0

Share this post


Link to post
Share on other sites
  • 0

Posted · Report post

It's lower than that.

0

Share this post


Link to post
Share on other sites
  • 0

Posted · Report post

Distribution of areas of 50,000 random triangles inside a circle.

post-1048-0-55681500-1395012356_thumb.gi

0

Share this post


Link to post
Share on other sites
  • 0

Posted · Report post

Amazingly, perhaps, the mode is zero.

Philosophically interesting: when sorted by size, more random triangles have zero area than any other size.

But that demands the question: how many triangles have zero area?

Why none, of course.

The probability of randomly choosing three coincident or collinear points is zero!

Pragmatically speaking, if we partition the possible area sizes into intervals, no matter how small,

and for this, see the previous post, the lowest-value interval will always characterize the greatest

number of random triangles.

0

Share this post


Link to post
Share on other sites

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!


Register a new account

Sign in

Already have an account? Sign in here.


Sign In Now

  • Recently Browsing   0 members

    No registered users viewing this page.