wu :: forums « wu :: forums - Area of the overlap » Welcome, Guest. Please Login or Register. Sep 16th, 2024, 6:46am RIDDLES SITE WRITE MATH! Home Help Search Members Login Register
 wu :: forums    riddles    medium (Moderators: william wu, SMQ, Eigenray, Icarus, Grimbal, towr, ThudnBlunder)    Area of the overlap « Previous topic | Next topic »
 Pages: 1 Reply Notify of replies Send Topic Print
 Author Topic: Area of the overlap  (Read 1395 times)
Junior Member

Posts: 57
 Area of the overlap   « on: May 23rd, 2014, 3:13pm » Quote Modify

Suppose you have a unit circle. You cut the circle into four equal parts but cutting vertically and horizontally through the diameter (perpendicular through the origin). Putting one of the four pieces on each of the following coordinates, (0,0) , (0,1) , (1,1) , (1,0) and fitting them together so that the radii produce a square causes the interior to have overlapping arcs. There is a piece, in the innermost center, where all four arcs overlap. Without using calculus, what is the area of the overlapped center?
 IP Logged
Junior Member

Posts: 57
 Re: Area of the overlap   « Reply #1 on: May 31st, 2014, 9:07pm » Quote Modify

Bump
 IP Logged
dudiobugtron
Uberpuzzler

Posts: 735
 Re: Area of the overlap   « Reply #2 on: Jun 1st, 2014, 4:05am » Quote Modify

Here's an outline of how I would start to approach it.  I don't actually have time to investigate it further now, though:

The corners of the middle shape are all findable using pythagoras, since they are on a right triangle with base = 0.5 and hypotenuse = 1.
(so they are each sqrt(3)/2 away from their opposite side).

The trick for finding the areas of the overlapping bits is to consider them as being made up of two half-segments.  To work out the area of a segment, work out the area of the sector it is part of, then subtract the inner triangle.

You can find the angles using trig since you know all of the sides.

Once you know those areas, it should be relatively easy to work out what to subtract.
 IP Logged
Uberpuzzler

Posts: 1029
 Re: Area of the overlap   « Reply #3 on: Jun 1st, 2014, 8:47am » Quote Modify

I apologize if I am missing some piece of common knowledge but I don't understand the meaning of "Bump".

Could anyone please explain? What does it mean?

 IP Logged
Junior Member

Posts: 57
 Re: Area of the overlap   « Reply #4 on: Jun 1st, 2014, 8:52am » Quote Modify

I moved the unanswered question back to the top of the forum by bumping it up
 IP Logged
Grimbal
wu::riddles Moderator
Uberpuzzler

Gender:
Posts: 7527
 Re: Area of the overlap   « Reply #5 on: Jun 1st, 2014, 2:59pm » Quote Modify

Without using calculus:
pi/3 - sqrt(3) + 1

The idea is to cut the pizza in 12 and keeping only the middle 1/3 of each quarter.  It does all the arcs.  You have some extra surfact to remove, but these are 8 identical triangles, easy to compute.
 IP Logged
 Pages: 1 Reply Notify of replies Send Topic Print

 Forum Jump: ----------------------------- riddles -----------------------------  - easy => medium   - hard   - what am i   - what happened   - microsoft   - cs   - putnam exam (pure math)   - suggestions, help, and FAQ   - general problem-solving / chatting / whatever ----------------------------- general -----------------------------  - guestbook   - truth   - complex analysis   - wanted   - psychology   - chinese « Previous topic | Next topic »

Powered by YaBB 1 Gold - SP 1.4!
Forum software copyright © 2000-2004 Yet another Bulletin Board