Ponder This Challenge - October 2010 - Finding the size of a pentagon

Ponder This Challenge:

In the following story, can you fill in the values a, b, c, d,and e with integer numbers so as to make the story true? And what would be X?

Ponder: What is the area of the lake?
This: I can not tell since I do not know anything about it.
Ponder: It has the shape of a convex pentagon; its edges lengths (in kilometers) are integers.
This: Can you give me more details?
Ponder: Here are the integer lengths of the edges: a, b, c, d,and e. I sorted them for you: a<b<c<d<e.
This: Now we are getting somewhere, but I still do not know the area.
Ponder: Last hint: Two of the angles are right (90 degrees).
This: Thank you. The lake area MUST be X square kilometers.

Update: October 4th: X, the area of the lake, does not have to be an integer.

Update: October 5th: Please specify, in your solution, the order of the edges.

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Solution

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  There are 120 (five factorial) ways for the order of the edges and two ways to place the right angles (either adjacent or not).

  Once we define the order and place the right angles, we can compute the area using the Pythagorean theorem and Heron's formula.
  Enumerating over several possible 5-tuples and reviewing all these possibilities for the lake, we can verify that there is only one possible area for a convex pentagon.

  The smallest solution we found is 2=4=12-6-5 (where "=" is a right angle) with an area of ~33.99.

  The smallest lake we know of (thanks, John T. Robinson ) that has an integer for its area is 2=5=20-6-13 (or 6=5=20-2-13) which has an area of 70.
  The 22=19=81-37-25 lake has a nice round area: 1000.

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  If you have any problems you think we might enjoy, please send them in. All replies should be sent to: ponder@il.ibm.com

Solvers

• Mark Perkins (10/01/2010 06:48 PM EDT)
• Chuck Carroll (10/02/2010 02:30 AM EDT)
• János Kramár (10/02/2010 03:57 AM EDT)
• Vladimir Sedach (10/03/2010 10:14 AM EDT)
• Michael Brand (10/02/2010 04:30 PM EDT)
• Daniel Bitin (10/03/2010 08:06 AM EDT)
• Singularity Institute (10/03/2010 06:25 PM EDT)
• Adam and Lael Daire (10/04/2010 04:44 PM EDT)
• Joseph DeVincentis (10/04/2010 04:59 PM EDT)
• Jason Crease (10/05/2010 12:23 PM EDT)
• John Tromp (10/05/2010 02:40 PM EDT)
• David Cole (10/05/2010 03:12 PM EDT)
• John T. Robinson (10/05/2010 08:38 PM EDT)
• Piet Orye (10/09/2010 12:48 PM EDT)
• Todd Will (10/10/2010 03:09 PM EDT)
• Albert Stadler (10/12/2010 01:12 PM EDT)
• Donald T. Dodson (10/14/2010 06:34 PM EDT)
• David Friedman (10/14/2010 07:28 PM EDT)
• Lei Liu (10/15/2010 05:32 AM EDT)
• Jeff Steele (10/16/2010 01:04 PM EDT)
• John G. Fletcher (10/17/2010 09:11 PM EDT)
• Liubing Yu (10/20/2010 06:35 AM EDT)
• Hamidreza Bidar (10/21/2010 09:00 AM EDT)
• Chris Shannon (10/21/2010 12:03 AM EDT)
• Dan Dima (10/21/2010 02:58 PM EDT)
• Jan Fricke (10/22/2010 09:50 AM EDT)
• Sylvain Becker (10/25/2010 11:16 AM EDT)
• Gale W. Greenlee (10/25/2010 12:54 PM EDT)
• Brian Kemper (10/28/2010 09:47 PM EDT)
• Qian Yang (10/31/2010 02:03 AM EDT)
• Phil Benamy (10/31/2010 06:51 PM EDT)