All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes > Geometry
Convex Hull (Posted on 2014-09-10) Difficulty: 3 of 5

  
Let ABCD be a cyclic quadrilateral with side lengths a, b, c, and d.

What is the area ( in terms of a, b, c, and d ) of the convex hull of
four circles ( with centers A, B, C, and D and each passing through
the circumcenter of ABCD )?
  

See The Solution Submitted by Bractals    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution Most of the answer Comment 1 of 1
The figure can be broken down into 9 separate pieces:
The quadrilateral itself (call this area X),
Sectors of the four circles with radius equal to that of the circumcircle of ABCD (call this radius r),
and four rectangles with areas ar, br, cr, dr.

The four sectors together make a complete circle.

Without finding X and r we can express the area as

X + pi*rē+ r(a+b+c+d)

The formulas of X and r can be found at http://mathworld.wolfram.com/CyclicQuadrilateral.html (4) and (8)


  Posted by Jer on 2014-09-11 11:56:14
Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (10)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information