@Cshearer41  
The red line, of length 2, is perpendicular to the bases of the three semicircles. What’s the total shaded area? pic.twitter.com/BVfUUlqyaB



Bilal
@Gogoljecco

Oct 23  
Catriona Shearer
@Cshearer41

Oct 23  
So close! But don’t forget they’re semicircles.


Professor Smudge
@ProfSmudge

Oct 23  
This is an intriguing question. You'd think the area might depend on the location of the red line. But if it doesn't, then the lefthand figure suggests the area is halfpi2squared minus pi1squared = pi pic.twitter.com/838j3C490Y


Catriona Shearer
@Cshearer41

Oct 23  
That was my thought too  how can it be enough information?! I’m waiting for someone to come up with a nice geogebra animation showing all the possible ways it can be drawn 😁


Dr M Thornber
@DrMThornber

Oct 23  
It’s a pity we don’t teach the intersecting chords theorem any more!


Catriona Shearer
@Cshearer41

Oct 23  
I learned the intersecting chords theorem via twitter a couple of months ago. Now I look for any excuse to use it 😁


Swaraj Kumar
@swarajkumar224

Oct 23  
Checked other responses, I always make the silly mistakes.


Catriona Shearer
@Cshearer41

Oct 23  
That one seems to have caught quite a few people out today!


kyle
@kyle_FD24

Oct 23  
It’s impossible.


Andreas Steiger
@mittelwertsatz

Oct 23  
Geogebra here:
geogebra.org/m/wxzhcuct


Catriona Shearer
@Cshearer41

Oct 23  
Amazingly, it is possible. The length of that line is enough to calculate the whole area, although it took me a while to convince myself it would work!


Catriona Shearer
@Cshearer41

Oct 23  
Thank you! 😁


Alex Cutbill
@intersectarian

Oct 23  
For those who are interested, the shaded region is an 'arbelos': en.wikipedia.org/wiki/Arbelos


Catriona Shearer
@Cshearer41

Oct 23  
Thanks! I was sure there would be a name for it  it seems far too nice a result not to have one  but it's the kind of thing that's quite hard to search for when you start with a picture. There are some other interesting results on that page, too.


Allymai
@Allymai_

Oct 23  
If you are allowed to assume that:
The radius of the larger semicircle must be 2, which means its area is 2π.
The radi of the smaller semicircles must be 1, which means that their areas are π/2 each, so their areas add to π.
Then the shaded area must be 2ππ=π.


Catriona Shearer
@Cshearer41

Oct 23  
Nice use of an assumption to simplify it 😄 but now, can you convince yourself that the area won't change as the smaller semicircles change size?


Allymai
@Allymai_

Oct 23  
Do I need to treat it as a more general thing?


david
@david_cobac

Oct 23  
Catriona Shearer
@Cshearer41

Oct 23  
The question kind of implied that the area would be there same whatever, so I’d say you were just taking an efficient short cut! But it’s interesting to work out why the area doesn’t depend on the relative sizes of the semicircles.

