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RE: Here is an old geometry/measurement chestnut for you.
November 8, 2018 at 2:45 am
(This post was last modified: November 8, 2018 at 2:51 am by Whateverist.)
(November 8, 2018 at 12:17 am)Fireball Wrote: My take-
Not convinced that point "D" is required.
Yeah, I think you're right. When I was writing it (I didn't do a first draft) I thought I might want it to refer to the part I wanted the area of but then I realized the "part of the second circle not inside the quarter circle" would get it done.
Edited to say I think there is an issue with your final answer but the wifey just got back from her trip so I can't figure out what it is right now, but I bet you can.
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RE: Here is an old geometry/measurement chestnut for you.
November 8, 2018 at 8:42 am
(This post was last modified: November 8, 2018 at 8:47 am by GrandizerII.)
D'oh, I did the same mistake as another poster here. Yeah, polymath's answer makes sense.
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RE: Here is an old geometry/measurement chestnut for you.
November 8, 2018 at 8:48 am
(November 8, 2018 at 2:45 am)Whateverist Wrote: (November 8, 2018 at 12:17 am)Fireball Wrote: My take-
Not convinced that point "D" is required.
Yeah, I think you're right. When I was writing it (I didn't do a first draft) I thought I might want it to refer to the part I wanted the area of but then I realized the "part of the second circle not inside the quarter circle" would get it done.
Edited to say I think there is an issue with your final answer but the wifey just got back from her trip so I can't figure out what it is right now, but I bet you can.
Yeah, I took the original diameter wrong. My answer should be twice what I originally came up with.
If you get to thinking you’re a person of some influence, try ordering somebody else’s dog around.
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RE: Here is an old geometry/measurement chestnut for you.
November 8, 2018 at 9:41 am
(November 7, 2018 at 11:26 pm)Whateverist Wrote: (November 7, 2018 at 9:08 pm)polymath257 Wrote: An old problem of Hipparchus.
Thanks for the Hipparchus reference. I didn't know that and will have to look into who that was. I don't recall where I encountered it but for whatever reason I woke up too early this morning thinking about it and had to recreate it.
I love the elegance of the solution though and how unlikely it seems when you consider the shape whose area we must find in itself. Very satisfying I think.
There is a generalization due to al-Haytham. Take a large half-circle with diameter AB. Inscribe a triangle ABC where C is on the half-circle. Now, draw two half circles with diameters AC and BC outside of the triangle.
The question concerns the area outside the original circle and inside the two smaller circles.
The Hipparchus figure is where the point C is on a perpendicular bisector of AB.
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RE: Here is an old geometry/measurement chestnut for you.
November 8, 2018 at 9:54 am
(November 7, 2018 at 6:43 pm)Whateverist Wrote: Draw a quarter circle and label the center point C and the points on either end of its arc, A and B.
Let the length of the quarter circle's radius = 1 unit
Now draw the circle with diameter AB, and label another point on this circle D such that point D is not inside the original quarter circle CAB.
Find the area of circle ABD outside of quarter circle CAB. Place answer inside hide tags and justify your result.
What's in it for me?
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RE: Here is an old geometry/measurement chestnut for you.
November 8, 2018 at 9:55 am
(November 8, 2018 at 9:54 am)Mathilda Wrote: (November 7, 2018 at 6:43 pm)Whateverist Wrote: Draw a quarter circle and label the center point C and the points on either end of its arc, A and B.
Let the length of the quarter circle's radius = 1 unit
Now draw the circle with diameter AB, and label another point on this circle D such that point D is not inside the original quarter circle CAB.
Find the area of circle ABD outside of quarter circle CAB. Place answer inside hide tags and justify your result.
What's in it for me?
Half a quarter of pie?
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RE: Here is an old geometry/measurement chestnut for you.
November 8, 2018 at 11:14 am
(This post was last modified: November 8, 2018 at 11:17 am by Whateverist.)
(November 8, 2018 at 9:41 am)polymath257 Wrote: (November 7, 2018 at 11:26 pm)Whateverist Wrote: Thanks for the Hipparchus reference. I didn't know that and will have to look into who that was. I don't recall where I encountered it but for whatever reason I woke up too early this morning thinking about it and had to recreate it.
I love the elegance of the solution though and how unlikely it seems when you consider the shape whose area we must find in itself. Very satisfying I think.
There is a generalization due to al-Haytham. Take a large half-circle with diameter AB. Inscribe a triangle ABC where C is on the half-circle. Now, draw two half circles with diameters AC and BC outside of the triangle.
The question concerns the area outside the original circle and inside the two smaller circles.
The Hipparchus figure is where the point C is on a perpendicular bisector of AB.
Oh goody. We're getting ready to go to the Y but I should have time for it in a few hours.
To really make the Hipparchus figure fit this generalization we just need to put two of the original quarter circles back to so that solution to that problem is double what we had before. That solution should then hold even if the half circles are not congruent.
First thoughts.
(November 8, 2018 at 8:48 am)Fireball Wrote: (November 8, 2018 at 2:45 am)Whateverist Wrote: Yeah, I think you're right. When I was writing it (I didn't do a first draft) I thought I might want it to refer to the part I wanted the area of but then I realized the "part of the second circle not inside the quarter circle" would get it done.
Edited to say I think there is an issue with your final answer but the wifey just got back from her trip so I can't figure out what it is right now, but I bet you can.
Yeah, I took the original diameter wrong. My answer should be twice what I originally came up with.
Yup.
(November 8, 2018 at 9:54 am)Mathilda Wrote: (November 7, 2018 at 6:43 pm)Whateverist Wrote: Draw a quarter circle and label the center point C and the points on either end of its arc, A and B.
Let the length of the quarter circle's radius = 1 unit
Now draw the circle with diameter AB, and label another point on this circle D such that point D is not inside the original quarter circle CAB.
Find the area of circle ABD outside of quarter circle CAB. Place answer inside hide tags and justify your result.
What's in it for me?
Well, a chestnut .. obviously. 
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RE: Here is an old geometry/measurement chestnut for you.
November 8, 2018 at 2:01 pm
(This post was last modified: November 8, 2018 at 2:07 pm by Whateverist.)
Well I misspoke in my first thoughts on the more general problem.
This more general problem would seem considerably more complex without having considered the easier form first.
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