Solved

# area of a snooker triangle

Posted on 2010-01-12
Medium Priority
429 Views
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
0
Question by:andieje
[X]
###### Welcome to Experts Exchange

Add your voice to the tech community where 5M+ people just like you are talking about what matters.

• Help others & share knowledge
• Earn cash & points
• Learn & ask questions
• 6
• 4
• 2
• +3
15 Comments

LVL 15

Expert Comment

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
0

LVL 84

Expert Comment

ID: 26295273
Do you mean 6  identical spheres?
Can you find the lengths of the sides of a triangle enclosing one sphere?
0

LVL 15

Accepted Solution

yuk99 earned 2000 total points
ID: 26295293
Sorry, it's (2sqrt(3)+4)R
0

LVL 2

Expert Comment

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
0

LVL 27

Expert Comment

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.
0

LVL 15

Expert Comment

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?
0

Author Comment

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!
0

LVL 15

Expert Comment

ID: 26296254
If you substitute R by 2 in my answer, you will get 14.928..., as you need.
0

Author Comment

ID: 26296275
Sorry, you are right. I can see my typo. It is 6 spheres
0

Author Comment

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
0

LVL 27

Expert Comment

ID: 26296307
You have already the complete answer in yuk99's response 10:35 and 10:38
0

LVL 15

Expert Comment

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.
0

LVL 15

Expert Comment

ID: 26296458
Mistyped: intersection of heights in equilateral triangle divides the height as 2:1
0

LVL 9

Expert Comment

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.
0

Author Closing Comment

ID: 31676210
thanks
0

## Featured Post

Question has a verified solution.

If you are experiencing a similar issue, please ask a related question

Lithium-ion batteries area cornerstone of today's portable electronic devices, and even though they are relied upon heavily, their chemistry and origin are not of common knowledge. This article is about a device on which every smartphone, laptop, anâ€¦
This article provides a brief introduction to tissue engineering, the process by which organs can be grown artificially. It covers the problems with organ transplants, the tissue engineering process, and the current successes and problems of the tecâ€¦
This is a video describing the growing solar energy use in Utah. This is a topic that greatly interests me and so I decided to produce a video about it.
I've attached the XLSM Excel spreadsheet I used in the video and also text files containing the macros used below. https://filedb.experts-exchange.com/incoming/2017/03_w12/1151775/Permutations.txt https://filedb.experts-exchange.com/incoming/201â€¦
###### Suggested Courses
Course of the Month12 days, 10 hours left to enroll

#### 777 members asked questions and received personalized solutions in the past 7 days.

Join the community of 500,000 technology professionals and ask your questions.