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area of a snooker triangle

Posted on 2010-01-12
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Last Modified: 2012-05-08
Hi

Another question:

A triangle frame is made to enclose 2 identical spheres. They are stacked in 3 rows (3,2,1). Each sphere has a radius of 2cm. Find the lengths of the sides of the frame.

I'm sure this is obivous, but again I can't see it :)

thanks
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Question by:andieje
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15 Comments
 
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Expert Comment

by:yuk99
ID: 26295271
May be 6 spheres?
You have 3 spheres on each side, it's 6R.
The distance from the center of corner sphere to triangle vertix is 2R, so the distance from vertix to projection of center to side can be calculated as sqrt(4R^2-R^2)=sqrt(3)R
So the side is (2sqrt(3)+6)R
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Expert Comment

by:ozo
ID: 26295273
Do you mean 6  identical spheres?
Can you find the lengths of the sides of a triangle enclosing one sphere?
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Accepted Solution

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yuk99 earned 2000 total points
ID: 26295293
Sorry, it's (2sqrt(3)+4)R
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LVL 2

Expert Comment

by:cenocre
ID: 26295400
You can get very close by just calculating the area of an equilateral triangle. The problem is that snooker triangles are rounded in the corners, thus cutting off a little of the area.

Given the radius of 2cm and 3 balls/side each side is 12cm

Plug it into this equation: Area= length of side squared x (square root of 3 / 4)

Area = 62.35
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LVL 27

Expert Comment

by:aburr
ID: 26295653
62.25 is too small for area. 12 cm is too small for side.
yuk99 has a very nice answer.
The tricky part is
"The distance from the center of corner sphere to triangle vertix is 2R"
but a good diagram will help derive it.
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Expert Comment

by:yuk99
ID: 26296157
To the author.
It's unclear do you need to calculate the area (in the title) or side length (in the question).
Is it ok to neglect roundness of real snooker triangle's corners?
Do we assume correctly that the question is about 6 spheres?
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Author Comment

by:andieje
ID: 26296213
hi,
i wrote the question verbatim. There is a picture with angluar corners and not round corners. There are 6 spheres. The answer is 14.9 if that helps you reverse engineer the question!
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Expert Comment

by:yuk99
ID: 26296254
If you substitute R by 2 in my answer, you will get 14.928..., as you need.
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Author Comment

by:andieje
ID: 26296275
Sorry, you are right. I can see my typo. It is 6 spheres
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Author Comment

by:andieje
ID: 26296288
I don't understand the answer. Not because it is poorly explained but because my geometry is poor. If you are willing to talk me through it, it is muh appreciated. Thanks
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LVL 27

Expert Comment

by:aburr
ID: 26296307
You have already the complete answer in yuk99's response 10:35 and 10:38
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LVL 15

Expert Comment

by:yuk99
ID: 26296433
1. Make a plot. You will see the side can be calculated as a sum of 4 sphere's radius and 2 equal segments from triangle corner to projection of center of sphere to the side. Is it clear.
2. How to calculate this segment? Consider the corner sphere as enclosed in a equilateral triangle. There is a theorem, that  divides intersection of heights in equilateral trianglehe height as 2:1. Try to prove it, it's easy. Smaller part of height is R, then larger part is 2R, the distance from sphere center to triangle corner.
3. Apply Pifagor theorem to calculate the projection: sqrt((2R)^2-R^2)=sqrt(4R^2-R^2)=sqrt(3)R
4. Finally just sum all segments.
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Expert Comment

by:yuk99
ID: 26296458
Mistyped: intersection of heights in equilateral triangle divides the height as 2:1
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Expert Comment

by:shadow77
ID: 26298857
Are you allowed to use trig?

Along one side of the triangle, drop perpendiculars from the centers of the first and third circles.  This will divide the side into three segments.

The middle segment will have a length of four radii (one each from the first and third circles and two from the center circle.  Hence, its length will be 4r = 8.

The outer segments will each have a length x = r / Tan(30 = pi/6) = 2 / 0.577350269189625 = 3.46410161513776.  This is true because we know from yesterday that a line from the center of the corner circle to the corner of the triangle bisects the corner angle (60).  Then, using trig, we see that Tan(30) = opposite side / adjacent side = r /x. or x = r/Tan(30).

The total length will be 8 + 2x = 14.9282032302755.
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Author Closing Comment

by:andieje
ID: 31676210
thanks
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